vi international conference - nsc.rumath.nsc.ru/lbrt/v1/conf2016/program.pdf · 2016. 8. 11. ·...

VI International Conference

Modern Problems in Theoretical and

Applied Probability

ProgrammeAbstracts

List of Participants

Novosibirsk 2016

SPONSORED BY:

• The Federal Agency for Scienti�c Organizations

• Russian Foundation for Basic Research

• Siberian Branch of Russian Academy of Science

• Sobolev Institute of Mathematics

• Novosibirsk State University

LOCATION:The Conference takes place at the Sobolev Institute of Mathematicslocated in Akademgorodok.Address: Acad. Koptyug prospect 4, 630090 Novosibirsk, Russia.

ORGANISING COMMITTEE:

• Vladimir LOTOV, Chair

• Mikhail FEDORUK, Rector of Novosibirsk State University

• Sergey GONCHAROV, Director of Sobolev Institute of Mathematics

• Alexander BYSTROV, Trip to Altai mountains

• Natalia CHERNOVA, Technical Co-Chair

• Sergey FOSS, Co-Chair, international contacts

• Alexander SAKHANENKO, Co-Chair, scienti�c programme

• Ekaterina SAVINKINA, Summer School

Modern Problems in Theoretical and Applied Probability

Programme

Monday, 22

9:00 � 9:30 Registration (Conference Hall)

9:30 � 10:00 Opening. 85th anniversary of AcademicianAlexander Alexeevich Borovkov (ConferenceHall)

Plenary Talks

10:00 � 10:45 Alexander Borovkov. Integro-local limittheorems for compound renewal processes.

14

10:50 � 11:35 Ildar Ibragimov. On a problem ofestimation of in�nite-dimensional parameterin lp spaces.

25

11:35 � 12:00 Co�ee Break

Early afternoon: Two Sessions in Parallel

Section �Limit Theorems� (Conference Hall)

12:00 � 12:30 Anatolii Mogulskii. Large deviationprinciples for trajectories of processes withindependent increments.

40

12:35 � 13:05 Vladimir Senatov. On the real accuracy ofapproximations in CLT.

51

Section �Stochastic Networks� (Room 417)

12:00 � 12:30 Peter M�orters. The contact process onevolving Scale-free networks.

40

12:35 � 13:05 Philippe Robert. On the dynamics ofrandom neuronal networks.

48

13:05 � 15:00 Lunch Time


Late afternoon: Two Sessions in Parallel

Section �Random Graphs� (Conference Hall)

15:00 � 15:30 Remco van der Hofstad. Rumor spreadand competition on scale-free random graphs.

23

15:35 � 16:05 Nelly Litvak. Dependencies and ranking indirected con�guration model.

34

Section �Selected topics� (Room 417)

15:00 � 15:30 V.V. Ulyanov, A.A. Lipatiev. Onnon-asymptotic results for MANOVA teststatistics with high-dimensional data.

58

15:35 � 16:05 Shakir Formanov. The Stein-Tikhomirovmethod and nonclassical CLT.

19

16:05 � 16:30 Co�ee Break

Evening: Two Sessions in Parallel

Section �Stochastic processes� (Conference Hall)

16:30 � 17:00 Thomas Mikosch. The auto- and cross-distance correlation functions of a multivariatetime series and their sample versions.

38

17:05 � 17:35 Mikhail Lifshits. Energy savingapproximations of random processes.

32

Section �Stochastic Geometry� (Room 417)

16:30 � 17:00 Hermann Thorisson. Mass-Stationarity,Shift-Coupling, and Brownian Motion.

54

17:05 � 17:35 G�unter Last. How to �nd an extra excursion. 31

18:00 � 20:00 Welcome Party


Tuesday, 23

Morning: Two Sessions in Parallel

Section �Queueing Theory� (Conference Hall)

10:00 � 10:30 Onno Boxma. Shot-Noise Processes inQueueing Theory.

16

10:35 � 11:05 Masakiyo Miyazawa. How parallel queuesare balanced for a join shortest queue anddedicated arrivals in heavy tra�c.

39

11:10 � 11:40 Guodong Pang. The Method of Chaining forMany Server Queues.

44

Section �Selected Topics� (Room 417)

10:00 � 10:30 Konstantin Borovkov. Approximatingwelfare in large e�cient markets.

15

10:35 � 11:05 C. Graham, F. Olmos, A. Simonian.Modeling and performance evaluation for theLeast Recently Used cache eviction policy.

21

11:10 � 11:40 Takis Konstantopoulos. Finite and in�niteexchangeability.

29

11:40 � 12:00 Co�ee Break


Section �Random Walks� (Conference Hall)

12:00 � 12:30 Valery Afanasyev. About time of reachinga high level by a random walk in a randomenvironment.

11

12:35 � 13:05 Thomas Mountford. Dynamic randomwalks with contact process environment.

42


Section �Skorohod Maps� (Room 417)

12:00 � 12:30 O�er Kella.On integral representation of theSkorohod map.

29

12:35 � 13:05 Haya Kaspi. An in�nite-DimensionalSkorohod Map and Continuous ParameterPriorities.

28

13:05 � 15:00 Lunch Time


Section �Mean-Field Models� (Conference Hall)

15:00 � 15:30 Kavita Ramanan. Large Deviations ofFinite-State Mean-Field Interacting.

46

15:35 � 16:05 Alexander Rybko. Mean-Field Limits ofLarge Queueing Networks.

48

Section �Statistics� (Room 417)

15:00 � 15:30 I.S. Borisov, Yu.Yu. Linke, P.S.Ruzankin. Uniform consistency of someclass of kernel estimators in nonparametricregression models.

13

15:35 � 16:05 Yuliana Linke. Asymptotic Properties ofOne-Step M-Estimators with Application toNonlinear Regression.

33

16:05 � 16:30 Co�ee break

Evening Poster session (Conference Hall)

16:30 � 17:15 Short 3-minute presentations

17:15 � 18:45 Presentation of posters


Poster presentations

Mikhail Chebunin. Stability of a Random Multiple AccessChannel with �Success-Failure� Feedback.

17

Anzhelika Kalenchuk. On explicit asymptotically normalestimators of an unknown parameter in a logarithmicregression problem.

27

Vyacheslav Lugavov. On the local limit theorem for sumsof random integer-valued variables de�ned on a Markov chaintransitions.

36

Alexey Medvedev. Speeding up non-Markovian FirstPassage Percolation by a single extra edge.

37

V.E. Mosyagin, N.A. Shvemler. Distribution of the timeof attaining the maximum for a Poisson process with negativedrift.

41

Ekaterina Savinkina. Construction and analysis of explicitestimators of unknown parameter in a power regressionmodel.

50

Olga Sukhovershina. On approximation in Koul's theoremfor weighted empirical processes.

53

Evgeniya Tsebukhovskaya. Construction andinvestigation of statistical estimators for pressure-derivatives.

57

Valentin Topchii, Vladimir Vatutin. Moments formultidimensional critical Bellman-Harris processes whereparticles life-lengths have in�nite mean and regularly varyingtails of di�erent order.

59

Maria Veretennikova. Controlled continuous time randomwalks and their extensions.

62

Galina Zverkina. On the coupling method for non-discreterenewal and regenerative processes.

66


Wednesday, 24

Morning: Two Sessions in Parallel


10:00 � 10:30 Dmitry Korshunov. Stability andinstability of di�usion processes and Markovchains with asymptotically zero drift.

30

10:35 � 11:05 Vitali Wachtel. First passage times forrandom walks with non-identically distributedincrements.

62

11:10 � 11:40 Alexander Sakhanenko. Structure andlimiting behavior of conditional random walkswith bounded local times.

49

Section �Di�usion Processes� (Room 417)

10:00 � 10:30 Natalia Smorodina. Analytic di�usionprocesses: de�nition, properties, limittheorems.

52

10:35 � 11:05 E. Presman, A. Slastnikov. Generalregular one-dimensional di�usion revisited.

45

11:10 � 11:40 A.Yu. Veretennikov, S.V. Anulova, H.Mai. On averaged control for 1D ergodicdi�usion.

60

11:40 � 12:00 Co�ee Break


Section �Selected Problems� (Conference Hall)

12:00 � 12:30 Andrey Zaitsev. Arak Inequalities forConcentration Functions and the Littlewood�O�ord Problem.

63

12:35 � 13:05 Valentin Topchii. Properties of renewalmatrices based on distribution functions withregularly varying tails.

55


Section �Random Walks� (Room 417)

12:00 � 12:30 Dmitrii Zaporozhets. Convex hulls ofrandom walks, hyperplane arrangements, andWeyl chambers.

64

12:35 � 13:05 Denis Denisov. Heavy-tra�c and heavy tailsfor random walks.

18

13:05 � 15:00 Lunch Time



15:00 � 15:30 Anton Tarasenko. Inequalities for thesojourn time of random walk above a certainboundary.

54

15:35 � 16:05 Vladimir Lotov. On the stationarydistribution of a semi-Markov randomwalk.

35

Section �Spatial Processes� (Room 417)

15:00 � 15:30 Sergei Zuyev. Segment recombinations andrandom sharing models.

65

15:35 � 16:05 Sergey Foss. Shape Theorems for Poissonhail on a bivariate ground.

20

Thursday, 25

Morning: Section �Selected Topics� (Conference Hall)

10:00 � 10:30 M. Remerova, M. Mandjes. An M/M/∞-type model for synchronization in the Bitcoinnetworks.

47

10:35 � 11:05 Farit Nasyrov. Integration of Systems ofStochastic Di�erential Equations.

43

Plenary Talks

11:10 � 11:20 Closing (Conference Hall)

11:20 � 12:00 Co�ee Break


Abstracts

About time of reaching a high level by arandom walk in a random environment

Afanasyev V.I.Steklov Institute, Moscow, Russia

Email: [email protected]

Let {Xn, n ≥ 0} be a random walk in a random environment (RWRE)and the random environment is a sequence of independent and identicallydistributed random vectors (pi, qi), i ∈ Z, where p0+q0 = 1, p0 > 0, q0 > 0.By de�nition it means that for a �xed random environment the sequence{Xn, n ≥ 0} is a discrete Markov chain starting from 0 with the set ofstates Z and transition probabilities pij such that pi,i+1 = pi, pi,i−1 = qi,i ∈ Z.

Let Tn = min {k ≥ 1 : Xk = n}, n ∈ Z, and κ = ln (q0/p0).In the case when a RWRE tends to +∞ (Eκ < 0), limit theorems for

Tn were established by Spitzer, Kozlov M.V. and Kesten.In the case of an oscillating RWRE (Eκ = 0, Eκ2 := σ2 ∈ (0,+∞)) the

author proved the following two results (see [1], [2]). Let {W (t) , t ≥ 0} bea standard Brownian motion. Set for t ∈ [0, 1]

Tt = inf {s ≥ 1 : W (s) = W (t)} , V (t) = sups∈[t,Tt]

|W (s)−W (t)| .

Then as n→∞{lnTbntc

σ√n, t ∈ [0, 1]

}D→ {V (1− t) , t ∈ [0, 1]}

(here we mean convergence in distribution in the space D [0, 1]);{lnTnσ√n

∣∣∣∣ Tn > T0

}D→ sup

0≤t≤1|W0 (t)| ,

where {W0 (t)} is a Brownian bridge.In the case when a RWRE tends to −∞ (Eκ > 0, E exp (−κ) <

+∞) there are the three di�erent types of RWRE in dependence on


the sign of Eκ exp (−κ). The author considered two of them (see [3]): ifEκ exp (−κ) = 0 then as n→∞{

lnTnσ√n

∣∣∣∣ Tn < +∞}

D→ sup0≤t≤1

|W0 (t)| ;

if Eκ exp (−κ) < 0 then

{lnTnσ√n

∣∣∣∣ Tn < +∞}

D→ sup0≤t≤1

W+0 (t) ,

where{W+

0 (t)}is a Brownian excursion.

[1] Afanasyev V.I., �About time of reaching a high level by a randomwalk in a random environment�, Theory Probab. Appl., 57:4 (2013),547�567.

[2] Afanasyev V.I., �Conditional limit theorem for maximum of randomwalk in a random environment�, Theory Probab. Appl., 58:4 (2014),525�545.

[3] Afanasyev V.I., �About time of reaching a high level by a transitiverandom walk in a random environment�, Teoriya veroyatnostei i eeprimeneniya, 61:2 (2016).


Uniform consistency of some class of kernelestimators in nonparametric regression models

I.S. Borisov, Yu.Yu. Linke, P.S. RuzankinSobolev Institute of Mathematics of the Russian Academy of Sciences,

Novosibirsk State University

We study the following heteroscedastic nonparametric regression model:

Xk = f(zk) + ξk, k = 1, . . . , n, (1)

where the function f(z) ∈ C[0, 1] is unknown, the design points {zk; k =1, . . . , n} are observable random variables, not necessarily identicallydistributed or independent, with unknown distributions on [0, 1]. Therefore,Model (1) with a �xed (nonrandom) design is a particular case of our study.Introduce the variational series 0 = zn:0 ≤ zn:1 ≤ . . . zn:n ≤ zn:n+1 = 1based on the sample {zk; k = 1, . . . , n}. The main assumption on the designis

limn→∞

max1≤i≤n+1

(zn:i − zn:i−1) = 0 a.s. (2)

It is clear that the limit relation (2) is valid in the case of a generalizedequidistant �xed design where zn:i := g(i/n) + o(1/n) uniformly in i,with a continuous function g. Moreover, if the random variables {zi} areidentically distributed and the interval [0, 1] is the support of their commondistribution then, under the strong mixing condition, the relation (2) isvalid as well.

Next we assume that the errors {ξk} which are not necessarilyindependent and identically distributed, satisfy with probability 1 thefollowing conditions:

E{zi}ξk = 0, supk

E{zi}ξ2k ≤ σ2, E{zi}ξkξm = 0

for all k,m ≤ n, k 6= m, where the symbol E{zi} denotes the conditionalexpectation given {zi; i = 1, . . . , n}, and σ2 is an unknown positiveconstant.

We suggest a convolution type estimator for f(z) in Model (1). Inparticular, for any precompact subset F of the Banach space C[0, 1] we


construct an estimator f∗F,n(z) such that, under condition (2),

supf∈F

sup0≤z≤1

|f∗F,n(z)− f(z)| p→ 0 as n→∞.

Integro-local limit theorems for compoundrenewal processes

A.A. Borovkov

Let (τ, ζ), (τ1, ζ1), (τ2, ζ2), . . . be a sequence of i.i.d. random vectors,

aτ := Eτ, aζ := Eζ, a :=aζaτ, σ2 :=

E(ζ − aτ)2

aτ,

Tn :=

n∑j=1

τj , Zn :=

n∑j=1

ζj , ν(t) := max{k : Tk < t}.

The compound renewal process (CRP) for the sequence (τj , ζj), j ≥ 1, isde�ned as

Z(t) := Zν(t), t > 0, Z(0) := 0.

We establish following integro-local theorem (ILT) in the range of normaldeviations.

Theorem. Let Eτ r <∞, E|ζ|r <∞ for some r ∈ (1 +√

2, 3] and theCram�er condition on the characteristic function of (τ, ζ) be met. Then, forany �xed ∆ > 0,

1

∆P(Z(t)− at ∈ [x, x+ ∆)

)=

1

σ√tφ

(x

σ√t

)+O(t−β), t→∞,

where φ is the standard normal density, β = r(r−1)2(r+1) >

12 , and the reminder

term O(t−β) is uniform in x and ∆ ∈ [t12−β , ct

3−r2 ).

The ILT for CRPs in the large deviation range is established as well.Let the Cram�er moment condition for (τ, ζ) be met and

A(λ, µ) := lnE exp(λτ + µζ).


Denote by λ = A(µ) the solution of the equation

A(−λ, µ) = 0.

Then, under some broad conditions, as t→∞ one has:

1. lnE exp(µZ(t)) = tA(µ) + o(t),

2.1

∆P(Z(t) ∈ [x, x+ ∆)

)=C(α)√

te−tD(α)

(1 + o(1)

), α :=

x

t,

whereD(α) := sup

µ

(αµ−A(µ)

),

∆ = ∆t → 0 slowly enough as t → ∞, and the reminder term o(1) isuniform in α ∈ K for some compact K. The function C(α) and admissiblecompacts K are described in explicit form.

Approximating welfare in large e�cientmarkets

K.A. Borovkov

We consider the e�cient outcome of a canonical economic trademarket. The model involves N buyers with unit demand and i.i.d. randomvaluations, and M sellers with unit supply whose costs are also i.i.d.random variables independent of the valuations. We approximate the jointdistribution of the quantityKα of units traded and the gainsWα from tradewhen there is a large number of market participants, i.e. both componentsof α := (N,M) tend to in�nity. The problem is reduced to studying aprocess expressed in terms of two independent empirical quantile processeswhich, in large markets, can be approximated by appropriately weightedindependent Brownian bridges. That allows to approximate the distributionof (Kα,Wα) by that of a functional of a Gaussian process. Moreover, wegive upper bounds for the approximation rate.

[Joint work with E.V. Muir]


Shot-Noise Processes in Queueing Theory

Onno Boxma

We consider shot-noise processes in relation to queueing theory. Inparticular, we consider a �uid network queueing system with service ratesthat have linear dependency on the workloads. These systems have anatural multidimensional shot-noise representation. We provide intuitionfor this representation by assuming a processor-sharing service discipline.We then formally prove the representation by making use of a level-crossing argument. Furthermore, for these systems we obtain a functionalcentral limit theorem, which turns out to be a multidimensional Ornstein-Uhlenbeck process by using an appropriate scaling of the arrival intensity.An artifact of the processor-sharing service discipline in case of linearservice rates, is that no customer ever leaves the system. This naturallyintroduces questions with respect to the number of customers in the systemif the service discipline is of another type, e.g. �rst-come-�rst-serve: in thiscase, new customers will increase the service rate of the customer underservice enough to complete its service within �nite time.In addition, it turns out that other problems are closely related. Inparticular, we study in�nite server systems with a Poisson parameterwhich is a stochastic shot-noise process (sometimes this type of intensityprocess is referred to as a Hawkes process) and we study shot-noiseprocesses driven by a shot-noise arrival process.

Note: This is joint work with David Koops and Michel Mandjes (Universityof Amsterdam) and Mayank (Eindhoven University of Technology).

[1] O. Kella and W. Whitt (1999). Linear stochastic �uid networks. J.Appl. Probab. 36, 244-260.


Stability of a Random Multiple AccessChannel with �Success-Failure� Feedback

Mikhail ChebuninNovosibirsk State University

We consider a decentralised multiple access model with an in�nitenumber of users, a single transmission channel, and an adaptivetransmission protocol that does not use the individual history of messages.There is no exchange of information between the users, and with any sucha protocol, each message present in the system in time slot [n, n + 1)is sent to the channel with probability pn that depends on the historyof feedback from the transmission channel, independently of everything else.

It is known since the 80's that with ternary feedback �Empty-Success-Collision� the channel capacity is e−1: if the input rate is below e−1,then there is a stable transmission protocol; and if the input rate isabove e−1, then any transmission protocol is unstable. By the ternaryfeedback, we mean the following: the users can observe the channeloutput and distinguish among three possible situations: either notransmission (�Empty�) or transmission from a single server (�Success�)or a collision of messages from two or more users (�Con�ict�). Similarresults (existence/nonexistence of a stable protocol if the input rate isbelow/above e−1) hold for systems with either of two binary feedbacks,�Empty-Nonempty� (a user cannot di�erentiate �Success� and �Failure�)or �Failure-Nonfailure� (a user cannot di�erentiate �Success� and �Empty�).

The third type of protocols, with �Success-Failure� binary feedback,is of a di�erent nature. Until recently, there were known only stableprotocols that use an extra information (testing package, individualhistory of messages etc.)

Foss, Hayek and Turlikov ([1]) introduced a new �doubly stochastic�protocol for stability of a system with the �Success-Failure� feedback.They showed that for any pair of numbers 0 < λ0 < λ1 < e−1, there existsa class of protocols that make stable a system with input rate λ, for anyλ ∈ [λ0, λ1]. However, any such a protocol depends explicitly on λ0 and


λ1.

In this paper, we prove existence of a stable �doubly stochastic�protocols that do no depend on value λ ∈ (0, e−1), and estimate therate of convergence to stationarity. A general form of such protocols wassuggested in [1], but without needed speci�cation of auxiliary functions.

[1] Sergey Foss, Bruce Hajek, Andrey Turlikov. Doubly RandomisedProtocols for a Random Multiple Access Channel with �Success-Nonsuccess� Feedback // Problems of Information Transmission,52(2):60-70, 2016.

Heavy-tra�c and heavy tails for random walks

Denis Denisov

Consider a family of random walks S(a)n with negative drift ES(a)

1 =

−a < 0 and �nite variance. Let M (a) = maxn≥0 S(a)n be the maximum of

the random walk.It is known that the probability P (Ma > x) decays exponentially fast

as a→ 0 (heavy tra�c asymptotics) and, for subexponential distributionsP (M (a) > x) decays according the the integrated tail as x → ∞. We willpresent a link between these two asymptotics and study the probabilityP (M (a) > x) and identify the regions of x for which the heavy tra�casymptotics and the heavy tail asymptotics hold.


The Stein-Tikhomirov method andnonclassical CLT

Shakir K. FormanovInstitute of Mathematics, National University of Uzbekistan


Suppose that F (x)is an arbitrary distribution function and

Φ(x) =1√2π

∫ x

−∞e−

u2

2 du

is a distribution function of the standard normal random variable (r.v.). In[1] Stein proposed an universal method for estimating the quantity

δ = supx|F (x)− Φ(x)|

We will modify Stein's method in terms of characteristic functions (ch.f.).Consider a class of ch.f.

F ={f(t) : f ′(0) = 0, σ2 = −f ′′(0) <∞

}and de�ne the Stein-Tikhomirov (S.-T.) operator

∆f(t) = f ′(t) + σ2tf(t).

LetXn1, Xn2, ..., n = 1, 2, ...

be an array of r.v.. Assume that

EXnj = 0, σ2nj = EX2

ni, j = 1, 2, ...,

∞∑j=1

σnj = 1

and denote

Sn = Xn1 +Xn2 + ... , Fn(x) = P (Sn < x) , n = 1, 2, ....

Theorem. The following convergence holds

supx|Fn(x)− Φ(x)| → 0, n→∞ ,


if and only if for any T > 0

sup|t|≤T

∑j

|∆ (fnj(t))| → 0 ,

where fnj(t) is the ch.f. corresponding to the distribution function Fnj(x).

[1] Stein placecountry-regionCh. A bound for the eror in thenormal approximation to the distribution of a sum of dependentrandom variables. Proceedings of the sixth Berkley Sympositionon Mathematical and Probability. PlaceTypeUniversity ofPlaceNameCalifornia Press, placeCityBerkeley (1972), pp. 583-602.

[2] Rotar V.I. On the generalization of the Lindeberg�Feller theorem.Mat.Zametki (Math. Notes). 18 (1), 129-135 (1975).

[3] Tikhomirov A.N. On the convergence rate in the CLT for weaklydependent random variables. Theory Probob. Appl. 25 (4), 800-818(1980).

Shape theorems for Poisson hail on a bivariateground

Sergey Foss

We consider the extension of the Euclidean stochastic geometry PoissonHail model to the case where the service speed is zero in some subset ofthe Euclidean space and in�nity in the complement. We use and developtools pertaining to subadditive ergodic theory in order to establish shapetheorems for the growth of the ice-heap under light tail assumptions on thehailstone characteristics. The asymptotic shape depends on the statisticsof the hailstones, the intensity of the underlying Poisson point process andon the geometrical properties of the zero speed set.

The talk is based on a joint paper with Francois Baccelli and HectorChang-Lara (Adv Applied Probab., 2016).


Modeling and performance evaluation for theLeast Recently Used cache eviction policy

Speaker: Carl Graham. Coauthors: Felipe Olmos, AlainSimonian

In order to save considerable time and network resources, cache serversplaced close to the users store a fraction of the dynamic catalog ofdocuments available from some central server. These caches are managedin real-time using algorithms called cache eviction policies. An importantquality of service (QoS) indicator is the hit probability, which is theprobability that a requested document is stored in the cache, and thecomplementary miss probability.

We present here the study [2] evaluating the miss probability for theLeast Recently Used (LRU) cache eviction policy. The cache is representedby a list of C ≥ 1 documents. Upon any user request for some documentin the catalog:

• If the document is already stored in the cache, then it is directlyuploaded to the user. Moreover, this document is moved to the frontof the list and all documents that were in front of it are shifted downby one slot.

• Else, the document is downloaded by the cache from the central serverand then uploaded to the user. Moreover, this document is placed atthe front of the list and all other documents are shifted down by oneslot except the last document which is eliminated from the cache.

We address the catalog and user preference dynamics using a Poissoncluster point process request model that has recently been independentlyproposed and studied heuristically in [3] and in [1]. We constructmathematically a probability space using Palm theory in order to considerand analyze a tagged document with respect to the rest of the requestprocess. This allows us to derive rigorously an integral formula for theexpected number of misses of the tagged document.

We obtain an expansion of the miss probability in powers of 1/C inthe natural scaling limit in which the cache size C and the arrival ratego to in�nity proportionally, using a probabilistic asymptotic analysis of


the integral formula. This expansion can be readily computed from thesystem parameters. It quanti�es and improves the accuracy of a widelyused heuristic called the Che approximation.

[1] Olmos, F., Kau�mann, B., Simonian, A., Carlinet, Y. (2014).Catalog dynamics: impact of content publishing and perishing onthe performance of a LRU cache. In 26th International Teletra�cCongress (ITC). IEEE, 1�9.

[2] Olmos, F., Graham, C., Simonian, A. (2015). Cache miss estimationfor non-stationary request processes. ArXiv:1511.07392.

[3] Traverso, S., Ahmed, M., Garetto, M., Giaccone, P., Leonardi, E.,Niccolini, S. (2013). Temporal locality in today's content caching:why it matters and how to model it. ACM SIGCOMM ComputerCommunication Review 43, 5�12.


Rumor spread and competition on scale-freerandom graphs

Remco van der HofstadDepartment of Mathematics and Computer Science, Eindhoven University

of TechnologyEmail: [email protected]

Real-world networks and models for them. Empirical �ndings haveshown that many real-world networks share fascinating features. Indeed,many real-world networks are small worlds, in the sense that typicaldistances are much smaller than the size of the network. Further, manyreal-world networks are scale-free in the sense that there is a high variabilityin the number of connections of the elements of the networks, makingthese networks highly inhomogeneous. Such networks are typically modeledusing random graphs with power-law degree sequences. In many real-worldnetworks, the degree power-law exponent τ is estimated to be in (2, 3), sothat the degrees are heavy-tailed with �nite mean yet in�nite variance.

A caricature model for such scale-free real-world networks is thecon�guration model, or the random graph with a prescribed degreesequence. In this model, we �x the number of vertices and denote this by n.We denote the vertices of the graph by [n] = {1, . . . , n}. For v ∈ [n], we letdv be its degree. In this talk, we assume that the degrees b = (dv)v∈[n] aregiven, and are such that their empirical distribution is close to power-lawdistributed, i.e.,

[1− Fn](x) =1

n

∑v∈[n]

1{dv>x} ≈ cx−(τ−1), (1)

where τ ∈ (2, 3) and, for an event E, 1E denotes its indicator. Here the totaldegree `n =

∑i∈[n] di, is assumed to be even. The CM on n vertices with

degree sequence d is constructed as follows: Start with n vertices and dvhalf-edges adjacent to vertex v ∈ [n]. Randomly choose pairs of half-edgesand match the chosen pairs together to form edges. Although self-loopsmay occur, these become rare as n→∞, thus giving a simple model thatis �exible in its degree structure.


Competition and rumor spread on scale-free random graphs. Inthis lecture, we investigate the behavior of competition processes on scale-free random graphs. Take two vertices uniformly at random, and place anindividual of two distinct types at these two vertices. Equip the edges withtraversal times, which could be di�erent for the two types. Then let each ofthe two types invade the graph, such that any other vertex will be occupiedby the type that gets there �rst.

We distinguish two cases. When the traversal times are exponential,we see that one (not necessarily the faster) type will occupy almost allvertices, while the losing type only occupies a bounded number of vertices,i.e., the winner takes it all, the loser's standing small. In particular, noasymptotic coexistence can occur. On the other hand, for deterministictraversal times, the fastest type always gets the majority of the vertices,while the other occupies a subpolynomial number. When the speeds arethe same, asymptotic coexistence (in the sense that both types occupy apositive proportion of the vertices) occurs with a positive probability thatwe can compute explicitly.

Rumor spread can be modeled in a very similar way by adding edge-traversal times on the edges of the graph. There, the main question is howfast a rumor from a single source invades the graph. A competition processis thus like the spread of a rumor up to the moment where the two speciesstart competing for territory.

Acknowledgements. This work is supported by the NetherlandsOrganisation for Scienti�c Research (NWO) through VICI grant639.033.806 and the Gravitation Networks grant 024.002.003. Jointwork with Enrico Baroni, Shankar Bhamidi, Mia Deijfen and GerardHooghiemstra.


On a problem of estimation ofin�nite-dimensional parameter in lp spaces

I. IbragimovSt. Petersburg Dept. Steklov Math. Institute Russian Ac. Sci.,

Math. Dept. of St. Petersburg State University

1. U. Grenander in his book "Abstract Inferences"( J. Wiley, 1981) hasconsidered the following non-parametric estimation problem.

We are observing a sample

X1, X2, . . . Xn

where the random variables Xj take integer values 1, 2, . . . with theprobabilities θ(k) = P{Xj = k}. The problem is to estimate the (in�nite-dimensional) parameter θ = (θ(1), θ(2), . . . .

In particular U. Grenander proved the following result

Theorem 1. If∑∞

1

√θ(j) < ∞, then the maximum likelihood estimates

(MLE) satisfy the following limit relation

limn→∞

√n||θ̂n − θ||1 = lim

∞∑1

√n|θ̂n(j)− θ(j)| =

√2

π

∞∑1

√θ(j)(1− θ(j)).

(1)

Here we continue the investigation of U. Grenander.2. The MLE θ̂n = (θ̂n(1), . . . ) where θ̂n(k) =

]{j,1≤j≤n:Xj=k}n . It follows

that with probability one√n(θ̂n− θ) ∈ lp, 1 ≤ p ≤ ∞. Denote Lp, 0<p<1

the metric space of x = (x1, . . . ),∑∞

1 |xj |p <∞ with the metric ρ(x, y) =∑∞1 |xj − yj |p; if

∑∞1 (θ(j))p < ∞, then

√n(θ̂n − θ) ∈ lp with probability

one.Denote Qn(p, θ) the distribution of the normed MLE

√n(θ̂n − θ) in

the space lp/ Denote Q(p, θ) the distribution in lp of the Gaussian randomsequence

η = (η1, . . . ) = (ξ1√θ(1)(1− θ(1)), . . . ξk

√θ(k)(1− θ(k), . . . )


whereξ1, ξ2, . . . are Gaussian random variables such that

Eξk = 0, Eξ2k = 1, Eξkξl = −

√θ(k)θ(l)

(1− θ(k))(1− θ(l).

Theorem 2. Let 0 < p ≤ ∞. If∑∞

1 (θ(k))p/2 <∞, then the distributionsQn(p, θ) converge to the distribution Q(p, θ), n → ∞. In other words forany real valued bounded continuous in lp function ϕ(x)

Eθϕ(√n(θ̂n − θ)) = Eϕ(η).

Theorem 3. Let l(x) ↑, x ∈ [0,∞). There exists a constant a > 0 such

that if l(x) ≤ const ·eax2

, then under the conditions of the previous theorem

limn

Eθl(√n(θ̂n − θ)||p) = El(||η||p), 1 ≤ p ≤ ∞. (2)

In particular ((cf (1))

limEθ||√n(θ̂n − θ)||pp =

1√2π

2p+12 Γ(

p+ 1

2)

∞∑1

(θ(j)(1− θ(j))p/2. (3)

The last equality is valid for all p > 0.

The MLE estimates are asymptotically e�cient in the sense ofI.Ibragimov, R. Khasminski , Statistical estimation: Asymptotic Theory,Springer, 1981.

It is a joint work with V. Ershov


On explicit asymptotically normal estimatorsof an unknown parameter in a logarithmic

regression problem1

Anzhelika KalenchukNovosibirsk State University, RussiaEmail: [email protected]

Let {Yi} be an observed sequence of random variables, {Xi} be knownconstants. Let they are connected by relations

Yi = ln(1 + αXi) + εi, i = 1, . . . , n,

where α is unknown an parameter, {εi} are unobservable independent andidentically distributed random variables. The research objective is to getexplicit estimate of the parameter α and investigate it.

Let us introduce the class of statistics:

α̂ =∑

i≤nbi∑

i≤naie

Yi

/∑i≤n

aiXi

∑i≤n

bieYi ,

where constants {ai} and {bi} satisfy the following conditions:∑i≤n

ai = 0,∑

i≤nbiXi = 0,

A =∑

i≤naiXi 6= 0, B =

∑i≤n

bi 6= 0.

Theorem 1. Let random variables {eYi(Bai−αAbi)} satisfy the Lindebergcondition and

∑i≤n b

2iDeYi/(

∑i≤n biEeYi)2 → 0. Then the statistic α̂ is

asymptotically normal estimator of the unknown α, i.e. as n→∞

(α̂− α)

dn⇒ N(0,1), for d2

n =Deεi

(Eeεi)2

∑i≤n

(1− αXi)2(aiA− αbi

B).

Therefore the explicit asymptotically normal estimator of an unknownparameter in a logarithmic regression problem is constructed. Earlier suchestimators were known only for three classes of non-linear regression: afractionally linear, a partially linear, and a power of degree 1/2.

The talk is based on a joint work [1] with A.I. Sakhanenko.1This work is supported by the RFBR-grant #15-01-07460a.


[1] A.A. Kalenchuk, A.I.Sakhanenko The existence of explicitasymptotically normal estimators of an unknown parameter ina logarithmic regression problem // Sib. Electr. Math. Reports, 12(2015), 874-883.

An in�nite-Dimensional Skorohod Map andContinuous Parameter Priorities

Haya Kaspi

Joint work with Rami Atar, Anup Biswas and Kavita RamananThe Skorokhod map on the half-line has proved to be a useful tool

for studying processes with non-negativity constraints. In this lectureI'll introduce a measure-valued analog of this map that transforms eachelement of a certain class of c�adl�ag paths that take values in the space ofsigned measures on the positive half line to a c�adl�ag path that takes valuesin the space of non-negative measures on that space. This is done in away that for each point x > 0, and a signed measure valued process (mt)the path t 7→ mt[0, x] is transformed via the classical Skorokhod map onthe half-line, and the regulating functions for di�erent x > 0 are coupled.We show that the map provides a convenient tool for studying queueingsystems in which tasks are prioritized according to a continuous parameter.Three such well known models are the EDF- earliest-deadline-�rst, theSJF- shortest-job-�rst and the SRPT- shortest-remaining-processing-timescheduling policies. Concentrating in this talk on the EDF, I'll show how themap provides a framework within which one forms �uid model equations,proves uniqueness of solutions to these equations and establishs convergenceof scaled state processes to the �uid model. In particular, for this model, ourapproach leads to new convergence results in time-inhomogeneous settings,which appear to fall outside the scope of existing approaches.


On integral representation of the Skorohodmap

O. Kella

A very short derivation of the integral representation of the two-sidedSkorokhod re�ection Z of a continuous function X of bounded variation ispresented, which is a generalization of the integral representation of theone-sided map featured in Anantharam and Konstantopoulos (2011). It isalso shown that Z satis�es a simpler integral representation when additionalconditions are imposed on X.

Finite and in�nite exchangeability

Takis KonstantopoulosDepartment of MathematicsUppsala University, Sweden

Exchangeability is ubiquitous in probability theory, with applicationsranging from statistical mechanics to stochastic networks and bayesianinference. The classical result in this area is de Finetti's theorem thatcompletely characterizes exchangeable probability measures on in�niteproducts of a "nice" space (e.g., a Polish space). But what happens toexchangeable measures on �nite products? It turns out that an analogousresult holds, but the mixing measure may not be positive. We shall presenta proof of this and show that no topological assumptions are neededwhatsoever. Subsequently, we ask the question of whether an exchangeablemeasure in n dimensions can be extended to n+1 or higher dimensions.(This is not always the case, and this is a problem that appears, e.g.,in extensions of statistical physics models to higher dimensions). Wegive a necessary and su�cient condition for this under some topologicalassumptions on the space. This is joint work with Svante Janson andLinglong Yuan.


Stability and instability of di�usion processesand Markov chains with asymptotically zero

drift

Dmitry KorshunovLancaster University, UK

In this talk we discuss transience, positive and null recurrence ofMarkov chains with asymptotically zero drift which is often referred to asLamperti's problem. Motivated by necessary and su�cient conditions fortransience, positive and null recurrence known for di�usions, we suggesttheir analogues for Markov chains. In this way we signi�cantly improvevarious conditions known in the literature for Markov chains.

Joint work with Denis Denisov (Manchester) and Vitali Wachtel(Augsburg).


How to �nd an extra excursion

G�unter LastKarlsruhe Institute of Technology

Liggett (2002) has posed and solved the following problem for a doublyin�nite sequence of independent and identically distributed coin tosses.Pick out a head in the sequence such that the distribution of the remainingcoin tosses (centred around the picked coin) is still that of the originalsequence. A similar question can be asked for a stationary Poisson processon the line (or on a general Euclidean space). Find a Poisson point such thatafter removing this point and centering the process around its position, theresulting process is still Poisson. Since the work of Holroyd and Peres (2005)it is well understood that these problems are closely related to invariantbalancing transports of random measures. In this talk we discuss ananalogous problem for a two-sided Brownian motion (Bt)t∈R. We considera set A of excursions with positive and �nite It�o measure and construct arandom time T ≥ 0 such the two-sided Brownian motion centered aroundT splits into three independent pieces: a time re�ected Brownian motionon (−∞, 0], an excursion distributed according to a conditional It�o law(given A), and a Brownian motion starting after this excursion. The proofrelies heavily on Palm and excursion theory and in particular on the resultspresented by Hermann Thorisson in his talk. This talk is based on joint workwith him and with Wenpin Tang (Berkeley).

[1] Holroyd, A.E. and Peres, Y. (2005). Extra heads and invariantallocations. Annals of Probability 33, 31�52.

[2] Last, G., Tang, W. and Thorisson, H. (2016). Transporting randommeasures on the line and embedding excursions into Brownianmotion. In preparation.

[3] Liggett, T.M. (2002). Tagged particle distributions or how to choosea head at random. In In and Out of Equlibrium (V. Sidoravicious,ed.) 133�162, Birkh�auser, Boston.


Energy saving approximations of randomprocesses2

M.A. LifshitsSt.Petersburg State University, Russia

MAI, Link�oping University.Email: [email protected]

Let (B(t))t∈Θ with Θ = Z or Θ = R be a wide sense stationaryprocess with discrete or continuous time. We investigate prediction andapproximation problems for B where optimization takes into accountother features of the objects we search for. One of the most motivatingexamples of this kind is an approximation of B by a stationary di�erentiableprocess X taking into account the kinetic energy that X spends in itsapproximation e�orts.

Consider H := span{B(t), t ∈ Θ} as a Hilbert space equipped with thescalar product (ξ, η) = E (ξη). For T ⊂ Θ let H(T ) := span{B(t), t ∈ T}.

Let L be a linear operator with values in H and de�ned on a linearsubspace D(L) ⊂ H. Consider the extremal problem

E |Y −B(0)|2 + E |L(Y )|2 → min, (1)

where the minimum is taken over all Y ∈ H(T )⋂D(L). The �rst term

in the sum describes approximation, prediction, or interpolation qualitywhile the second term stands for additional properties of the object we aresearching for, e.g. for the smoothness of the approximating process. Thisproblem includes (with L = 0) the classical prediction and interpolationproblems.

Recall the spectral representation B(t) =∫eituW(du) where W is

an orthogonal random measure with E |W(A)|2 = µ(A), µ being thespectral measure of B. The operators L we handle are L

(∫φ(u)W(du)

):=∫

`(u)φ(u)W(du). For non-adaptive approximation we show that the unique

solution of (1) exists and is given by Y =∫ (

1 + |`(u)|2)−1W(du) and the

minimum in (1) is equal to∫ |`(u)|2

1+|`(u)|2 µ(du). The same result holds for the

2This is a joint work with I.A. Ibragimov (PDMI, St.Petersburg) and Z. Kabluchko(M�unster University).


processes with stationary increments. We also prove appropriate extensionsof the classical Kolmogorov and Krein prediction singularity criteria andKolmogorov's criterion of error-free interpolation.

Asymptotic Properties of One-StepM-Estimators with Application to Nonlinear

Regression

Yuliana Yu. LinkeSobolev Institute of Mathematics, Russia


We study the asymptotic behavior of one-step M -estimators basedon independent not necessarily identically distributed observations. Theseestimators are usually regarded as explicit approximations of consistentM -estimators and generalize Fisher's one-step approximations of consistentmaximum likelihood estimators. In particulary, we �nd quite generalconditions for asymptotic normality of one-step M -estimators even if thecorresponding consistent M -estimators do not exist.

As an application, we consider the problem of �nding optimal (in somesense) estimators for nonlinear regression models. We also consider theproblem of constructing initial estimators in nonlinear regression modelswhich are needed for one-step estimation procedures.


Dependencies and ranking in directedcon�guration model

N.Litvak

A con�guration model is a natural and convenient null-model forcomplex networks because it preserves the degree distribution butintroduces a completely random wiring between the nodes. When wewant to understand how the network structure a�ects algorithms andmeasurements on real-life network, comparison to the con�guration modelis highly informative. In this talk we present some recent results on thedirected con�guration model. First, we address so-called structural negativedegree-degree correlations that usually arise under the assumption that thegraph is simple. We �nd the connection between the structural correlationsand the number of erased edges in the erased con�guration model. For thelatter, we establish new upper bounds in terms of the size of the graph.Next, we consider the distribution of a family of rankings, which includesGoogle's PageRank, on a directed con�guration model, and show that therank of a randomly chosen node in the graph converges in distribution toa �nite random variable that can be expressed through the endogenoussolution to a stochastic �xed-point equation.


On the stationary distribution of asemi-Markov random walk 3

V.I. LotovSobolev Institute of MathematicsNovosibirsk State UniversityEmail: [email protected]

We �nd the stationary distribution of a stochastic process with delayingscreen at origin. The trajectories of the process have linear growth betweenrandom jumps at some random times. The process has negative drift;it starts from random point every time after passing zero level. Someparticular versions of this process were considered in [1], [2]. We obtainrepresentations for the stationary distribution by combining asymptoticresults for regenerative processes and factorization technique in boundarycrossing problems for random walks.

[1] T.I. Nasirova, E.A Ibayev and T.A. Aliyeva. The Laplace transform ofthe ergodic distribution of the process semi-markovian random walkwith negative drift, nonnegative jumps, delays and delaying screen atzero. Theory of Stochastic Processes, 2009, Vol 15(31), no.1, pp.49-60.

[2] J. Yapar, S. Maden, U. Karimova. Laplace transform of thedistribution of the Semi-Markov walk process with a positive drift,negative jumps, and a delay screen at zero. Automatic Control andComputer Sciences, 2013, Vol. 47, No. 1, pp. 22-27.

3The author was partially supported by the Russian Foundation for Basic Research(Grant 16-01-00049).


On the local limit theorem for sums of randominteger-valued variables de�ned on a Markov

chain transitions

Vyacheslav S. Lugavov

Let κ = {κ(n);n = 0, 1, . . .} be an irreducible and aperiodic Markovchain with state space D = {1, 2, . . . , N} and with transition matrixP =‖ prj ‖r,j=1,N . For each possible transition (i, j), let Fij(x) be adistribution function corresponding to an integral random variable with a�nite second moment. Set S(0) = 0, S(n) = ξ1 +ξ2 + . . .+ξn, where ξm hasthe distribution Fij(x) if m-th transition takes the chain from state i to statej . Consider the two-dimensional Markov process L =(S(n), κ(n)), n ≥ 0.We suppose that the process L is not degenerate [1]. Then we set thelocal limit theorem for S(n). This theorem generalizes the correspondingresult from [2], because analytic condition (2.1) introduced by Miller is notassumed to be satis�ed.

[1] Keilson J.,Wishart D.M.G. A central limit theorem for processesde�ned on a �nite Markov chain // Proc. Cambridge Philos. Soc.1964. Vol. 60. P. 547-567.

[2] Miller H.D. A convexity property in the theory of random variablesde�ned on a �nite Markov chain // Ann. Math. Statist. 1961. Vol.32.P. 1260-1270.


Speeding up non-Markovian First PassagePercolation by a single extra edge

Alexey MedvedevCentral European University, Budapest, Hungary

Sobolev Institute of Mathematics, Novosibirsk, Russia

Gabor PeteAlfr�ed R�enyi Institute of Mathematics

Technical University of Budapest (BME), Budapest, Hungary

One model of real-life spreading processes is First Passage Percolation(also called SI model) on random graphs. Social interactions often followbursty patterns, which are usually modelled with non-Markovian heavy-tailed edge weights. On the other hand, random graphs are often locallytree-like, and spreading on trees is very slow, because of bottleneck edgeswith huge weights. We show the surprising phenomenon that adding asingle random edge to a tree typically accelerates the process severely. Weexamine this acceleration e�ect on some natural models of random trees:critical Galton-Watson trees conditioned to be large in some way, uniformrandom trees using Loop Erased Random Walks and P�olya urn ideas, andwill also discuss what should happen on near-critical Erd�os-R�enyi graphs.


The auto- and cross-distance correlationfunctions of a multivariate time series and

their sample versions

T. MikoschCopenhagen

This is joint work with R.A. Davis, P. Wan (Columbia Statistics), andM. Matsui (Nagoya).

Feuerverger (1993) and Sz�ekely, Rizzo and Bakirov (2007)introduced the notion of distance covariance/correlation as a measure ofindependence/dependence between two vectors of arbitrary dimension andprovided limit theory for the sample versions based on an iid sequence. Themain idea is to use characteristic functions to test for independence betweenvectors, using the standard property that the characteristic function of twoindependent vectors factorizes. Distance covariance is a weighted versionof the squared distance between the joint characteristic function of thevectors and the product of their marginal characteristic functions. Similarideas have been used in the literature for various purposes: goodnes-of-�ttests, change point detection, testing for independence of variables,... ; seework by Meintanis, Hu�skova, and many others. In contrast to Sz�ekely et al.who use a weight function which is in�nite on the axes, the latter authorschoose probability density weights. Z. Zhou (2012) extended distancecorrelation to time series models for testing dependence/independence ina time series at a given lag. He assumed a �physical dependence measure�.

In our work we consider the distance covariance/correlation for generalweight measures, �nite or in�nite on the axes or at the origin. These includethe choice of Sz�ekely et al., probability and various L�evy measures. Thesample versions of distance covariance/correlation are obtained by replacingthe characteristic functions by their sample versions. We show consistencyunder ergodicity and weak convergence to an unfamiliar limit distribution ofthe scaled auto- and cross-distance covariance/correlation functions understrong mixing. We also study the auto-distance correlation function of theresidual process of an autoregressive process. The limit theory is distinctfrom the corresponding theory of an iid noise process. We illustrate thetheory for simulated and real data examples.


How parallel queues are balanced for a joinshortest queue and dedicated arrivals in heavy

tra�c

Masakiyo MiyazawaTokyo University of Science

We consider d parallel queues with single servers, numbered as1, 2, . . . , d. There are d+1 arrival streams, numbered as 0, 1, 2, . . . , d, whichare independent renewal processes. Arriving customers in stream 0 choosethe shortest queue with tie break, while arriving customers in stream i 6= 0join queue i, which are called dedicated arrivals. In each queue, customersare �rst-come �rst-served with i.i.d. service times, which are independentof everything else. We refer to this queueing model as a generalized JSQ.

We are interested to see how queues in this model are balanced inthe steady state. For this, we consider the stationary joint queue lengthdistribution, but it is hard to analytically derive it. Thus, we consider itsapproximation in heavy tra�c. Let J = {1, 2, . . . , d}, and J+0 = {0} ∪ J .Let the mean arrival rate of stream i be λi for i ∈ J+0, and let the meanservice rate of service j be µj for j ∈ J . Then, the stability condition is∑i∈J+0

λi <∑i∈J µi and λj < µj for j ∈ J .

For heavy tra�c approximation, we consider a sequence of thegeneralized JSQ's, indexed by n = 1, 2, . . .. We denote its λi and µj by

λ(n)i and µ(n)

j . Let

r(n)0 =

∑i∈J

µi −∑i∈J+0

λi, r(n)j = µ

(n)j − λ(n)

j , j ∈ J,

and assume that, for each i ∈ J+0 and j ∈ J , r(n)i , λ

(n)i , µ

(n)j > 0 converge to

nonnegative constants ri, λi, µj , respectively, as n→∞. As usual, the 2ndmoments of the inter-arrival and service times are assumed to uniformlyconverge. Then, r0 = 0 is called a heavy tra�c condition. Under thiscondition, we scale the queue lengths of the n-th generalized JSQ by r(n)

0 ,and consider the weak limit of a sequence of their stationary distributions.We consider under what conditions which group of queues are well balanced,that is, their di�erence vanishes in the limiting distribution, which may beconsidered as state space collapse.


Reiman (1984) shows that |λ1 − λ2| < λ0 is su�cient for two queues tobe well balanced for d = 2 in the sense of process limit, but no necessarycondition is considered. We take a di�erent approach from Reiman's. Oursis based on a martingale representation of a piecewise deterministic Markovprocess, which has been studied in Miyazawa (2017) (see also Braverman,Dai and Miyazawa (2015)). We �rst consider the case of d = 2, for which wefully answer the problem. Then, we consider the case of general d, which ispartially answered. We also study the limiting distribution in heavy tra�cwhen some of queues are not well balanced.

Large deviation principles for trajectories ofprocesses with independent increments

Anatolii A. Mogulskii

The talk is devouted to the large deviation principles for processeswith independent increments. The results include the so-called local andextended large deviation principles that hold in those cases where the"usual"(classical) large deviation principle is unapplicable.

The contact process on evolving scale-freenetworks

Peter M�orters

We study the contact process on a class of scale-free networks, whereeach vertex updates its connections at independent random times. We showthat there is a phase transition between a phase where for all infection ratesthe infection survives for a long time and a phase where for su�cientlysmall infection rates extinction occurs quickly. The transition occurs whenthe power law exponent crosses the value four. This behaviour is di�erentfrom the behaviour of both static and mean �eld models. The talk is basedon joint work with Emmanuel Jacob (Lyon).


Distribution of the time of attainingthe maximum for a Poisson process

with negative drift

V.E. Mosyagin, N.A. ShvemlerTyumen State University, Tyumen, Russia

Let ν−(t), ν+(t) be independent standard Poisson processes extendedby zero for t > 0.

De�ne stochastic process:

Y (t) = at− ν+(pt) + ν−(−qt), t ∈ (−∞,∞). (1)

Inequalities p > a > q > 0 provide negative middle drift. The process (1)was determined in [1, p. 329] as the limit (in some sense) process of thelogarithm of the likelihood ratio for normalized argument in the case ofdiscontinuous density.

Let G(x) denote the d.f. of the argmax for the process (1).Theorem. The distribution G(x) is expressed in the form:

G(x) = β

ax∫0

e−pax

[z]∑k=0

(pa

)kψ (z − k)

(zk

k! −zk−1

(k−1)!

)dz+ (p−a)q

(p−q)a , x ≥ 0, (2)

where [2, p. 167]

ψ(z) = P

(supt<0

Y (t) ≤ z)

=(1− q

a

) [z]∑k=0

(−1)k(qa

)k (z−k)k

k! eqa (z−k),

and β > 0 is a unique solution of the equation(1− e−β

)/β = a/p.

Although these expressions are quite di�cult, the integral in (2) can befound successively at any intervals n− 1 ≤ ax < n, n = 1, 2, ...

For example, if 0 ≤ ax < 1, than

G(x) = (a−q)β(p−q)

(1− e−(p−q)x

)+ (p−a)q

(p−q)a .

[1] I. A. Ibragimov and R. Z. Khasminskii, Asymptotic Theory ofEstimation [in Russian], Nauka, Moscow (1979)


[2] A. V. Skorokhod, Stochastic Processes with Independent Increments[in Russian], Nauka, Moscow (1964)

Dynamic random walks with contact processenvironment

Thomas Mountford

In joint work with Eulalia Vares, we consider the model introduced byDen Hollander and dos Santos: A particle performs a random walk whosejump rates depend on the "environment that is the state of a supercriticalcontact process in a local fashion. We establish the law of large numbersand the invariance principle for all supercritical densities of the contactprocess.


Integration of Systems of StochasticDi�erential Equations

Nasyrov F. S.Ufa

Consider the following system of stochastic di�erential equationswith respect to d-mentional Brownian motion process W (s) ={W1(s), . . . ,Wd(s)} ηi(t) = η0

i +t∫

0

Bi(s, η(s),W (s))ds+d∑j=1

t∫0

σij(s, η(s)) ∗ dWj(s),

i = 1, 2, . . . , n, (1)

the stochastic integrals in the right parts of these equations are Strato-novich integrals.

In order to ensure the existens and uniqueness of a solution, we needto impose Lipschitz and the linear growth conditions. In addition, supposethat every column of matrix Σ = {σij} has the bounded away from zeroelement.

It is shown that ηi(t) = ϕi(t,W1(t), . . . ,Wd(t), C1(t), . . . , Cn(t)), i =1, . . . , n, where ϕi(t, w1, . . . , wd, c1, . . . , cn) are determinate functions,Ck(t), k = 1, 2, . . . , n, � adapted random processes. Moreover, the functionsϕi are solutions of the normal system of determinate ordinary di�erentialequations. On the other hand, the random functions Ck(t), k = 1, 2, . . . , n,are solutions of the normal system of stochastic ordinary di�erentialequations.

[1] Nasyrov F. S. Local Times, Symmetric Integrals and StochasticAnalysis. FIZMATLIT, 2011.


The Method of Chaining for Many ServerQueues

Guodong PangPenn State University

The method of chaining, originating from Kolmogorov, has been a verypowerful tool to obtain probability and moment bounds for stochasticprocesses. We explore the application of the method of chaining innon-Markovian many-server queues with a general arrival process, andwith either (i) general time-varying service times (e.g., arrival dependentservices), or (ii) weakly dependent service times. In these models, we studytwo-parameter stochastic processes that can be used to describe the systemdynamics, in particular, X(t, y) representing the number of jobs in thesystem at time t that have received an amount of service less than orequal to y (or that have a residual amount of service strictly greater thany). We prove functional central limit theorems for these two-parameterprocesses. The method of chaining provides important maximal probabilityand moment bounds on the two-parameter processes, which are key toprove their convergence. (This is joint work with Yuhang Zhou at PennState University.)


General regular one-dimensional di�usionrevisited

Ernst Presman, Alexander SlastnikovCEMI RAS, Moscow, Russia

A general one-dimensional di�usion is a homogeneous in time strongMarkov process Xt with continuous trajectories and values in a union ofan interval I and a �cemetery� e. Let τy, τe be the �rst hitting times ofthe state y ∈ I and the state e, respectively. The regularity means thatPx[τy < τe] > 0 for any x, y ∈ int I. Such di�usion is characterized bythree functions:

1) strictly increasing right-continuous m(x), which generates the speedmeasure M(dx), such that M(]a, b]) = m(b)−m(a);

2) strictly increasing continuous s(x) (scale), which generates the scalemeasure S(dx);

3) non-increasing right-continuous k(x), which generates the killingmeasure K(dx).

If a boundary point is accessible from inner point then for correctde�nition of di�usion one needs to de�ne the �nite or in�nite values ofmeasures M and K in the respective point.

If sequences of speed measures Mn(dx), scale measures Sn(dx) andkilling measures Kn(dx) weakly converge to speed measure M(dx), scalemeasure S(dx) and killing measure K(dx) then corresponding di�usionprocesses weakly converges to the di�usion with characteristics M(dx),S(dx) and K(dx).

Let G be the set of functions f , such that limv↓0

f(x+ v)− f(x)

s(x+ v)− s(x)=d+f

dS(x)

exists for all x ∈ int I and is a function of bounded variation on any interval[a, b[, a, b ∈ int I.

For each f ∈ G let us de�ne a signed measure Lf(dx) such that

Lf([a, x[) =d+f

dS(x)− d+f

dS(a)−

∫[a,x[

f(v)K(dv).

Let us denote: G+ = {f ∈ G : the measure Lf(dx) is nonnegative};


G− = {f ∈ G : Lf(dx) is nonpositive}; G0 = G+ ∩ G− = {f ∈ G :Lf(dx) ≡ 0}.

Statement 1. a) The set G coincides with the domain of the weakgenerating operator of the process Xt; b) G0 coincides with the set ofpotential functions; c) G− coincides with the set of excessive functions.

Since the process Xt (t ≥ 0) is strong markovian and continuous itfollows that there exist a continuous nondecreasing function h1(x) and acontinuous nonincreasing function h2(x), x ∈ int I such that

Px[τy <∞] =h1(x)

h1(y)for y > x, Px[τy <∞] =

h2(x)

h2(y)for y < x.

Statement 2. If there exist two points x, y ∈ I such that Px[τy <∞] < 1 then any function from G0 is a positive linear combination of h1(x)and h2(x).

The work was supported by RFBR (project 15-06-03723).

Large Deviations of Finite-State Mean-FieldInteracting Particle Systems

Kavita RamananBrown University, Division of Applied Mathematics

We establish a large deviation principle for the empirical measureprocess associated with a general class of �nite-state mean �eld interactingparticle systems with Lipschitz continuous transition rates that satisfya certain ergodicity condition.The main novelty is that more than oneparticle is allowed to change its state simultaneously, and so a standardapproach to the proof based on a change of measure with respect to asystem of independent particles is not possible. The result is shown tobe applicable to a wide range of models arising from statistical physics,queueing systems and communication networks.We will also establish acertain locally uniform re�nement of the large deviation principle anddescribe implications of this result for the stability of associated �uidmodels.This is joint work with Paul Dupuis and Wei Wu.


An M/M/∞-type model for synchronization inthe Bitcoin network

M. RemerovaKorteweg-de Vries Institute for Mathematics, University of Amsterdam,

The NetherlandsEmail: [email protected]

M. MandjesKorteweg-de Vries Institute for Mathematics, University of Amsterdam,CWI, Amsterdam, Eurandom, Eindhoven University of Technology, The

NetherlandsEmail: [email protected]

The model we present is inspired by the blockchain update process inthe Bitcoin network. It is a version of the M/M/∞ queue where customersdo not depart one-by-one but in batches of uniform size. We compare theconventional and the �bitcoin"versions of the M/M/∞ queue in a numberof aspects. The main result we discuss is the �uid limit approximation of thepopulation process in presence of service delays. That is, we let the servicerate µ → ∞. While the conventional M/M/∞ queue would require thespace-time scaling by µ and admit a deterministic �uid limit, the bitcoinmodel requires the space-time scaling by

√µ and has a random �uid limit,

which is a rare type of result.


On the dynamics of random neuronal networks

Philippe Robert

We study the mean-�eld limit and stationary distributions of a pulse-coupled network modeling the dynamics of a large neuronal assemblies.Our model takes into account explicitly the intrinsic randomness of�ring times, contrasting with the classical integrate-and-�re model. Theergodicity properties of the Markov process associated to �nite networksare investigated. We derive the limit in distribution of the sample path ofthe state of a neuron of the network when its size gets large. The invariantdistributions of this limiting stochastic process are analyzed as well astheir stability properties. We show that the system undergoes transitionsas a function of the averaged connectivity parameter, and can supporttrivial states (where the network activity dies out, which is also the uniquestationary state of �nite networks in some cases) and self-sustained activitywhen connectivity level is su�ciently large, both being possibly stable.

Joint work with Jonathan Touboul.

Mean-Field Limits of Large QueueingNetworks

Alexander Rybko

The investigation of the mean-�eld limit behaviour is a useful tool forstudying qualitative properties of queueing networks. We use this �uid limitto �nd nontrivial attractors of dynamical systems describing the evolutionof large symmetric networks.These nonlinear dynamical systems describethe asymptotic behaviour of large symmetric networks in the mean-�eldlimit, when number of nodes of symmetric networks tends to in�nity. Thestructure of attractors of such dynamical systems give us the possibilityto verify Poisson hypothesis. Of special interest is the mean-�eld limitof networks with moving servers. They have unexpected properties. Someexamples will be analyzed.


Structure and limiting behavior of conditionalrandom walks with bounded local times 4

Alexander I. SakhanenkoNovosibirsk State University

Sobolev Institute of Mathematics, Novosibirsk, RussiaEmail: [email protected]

We consider a random walk {Xt} on the integers Z = {0,±1,±2, ...}with transition probabilities pk = P(Xn+1−Xn = k) where 0 < p1 < 1 andpk = 0 for all k > 1. Further, assume each state j ∈ Z is initially given arandom units of energy and each time when the random walk visits a state,it takes 1 unit of energy to leave it. Eventually such a random walk arrives(with probability 1) to a state with no energy � then it "freezes"there.

For any state n > 0, let T (n) be the �rst time the random walk visits nand let T (n) be in�nite if n is never visited. Clearly, each T (n) takes in�nitevalue with a positive probability that tends to 1 as n tends to in�nity.

We consider the trajectory of {Xt} on the time interval 0, 1, ..., T (n)conditionally on the event that T (n) is �nite. We show that, as n tendsto in�nity, this trajectory converges in a "strong"sense to a trajectory of aregenerative process.

We also consider another properties of the model.This work is motivated by a paper by I. Benjamini and N. Berestycki

(2010) where the case of a simple symmetric random walk with non-randomconstrains has been considered.

The talk is based on a joint work with Sergey G. Foss (Heriot-WattUniversity, Edinburgh and Sobolev Institute of Mathematics, Novosibirsk).

[1] I. Benjamini, N. Berestycki. Random paths with bounded local time.J. Eur. Math. Soc., 12 (2010), 819-854.

4This work is supported by the RFBR-grants #14-01-00220a and #15-01-07460a.


Construction and analysis of explicitestimators of unknown parameter in a power

regression model5

Ekaterina SavinkinaNovosibirsk State University, Russia


The subject of my research work is to estimate an unknown parameterin a special nonlinear regression problem. Suppose we observe randomvariables {Yi} represented in a way

Yi =√

1 + αxi + εi, i = 1, 2, . . . ,

where {εi} are unobservable independent random variables and {xi >0} are some non-random observations. We require the following usualconditions to be met:

Eεi = 0, 0 < Var(εi) = σ2 <∞.Our aim is to estimate an unknown parameter α > 0. This problem

presents a standard example of nonlinear regression where ordinary leastsquares estimation faces some serious computing di�culties. During myresearch work it was found that explicit unbiased estimator of the unknownparameter can be represented as a ratio of two linear statistics dependingon specially chosen constant values {cni, n = 2, 3, . . . , i = 1, . . . , n}:

α∗n =

n∑i=1

cniY2i

/ n∑i=1

cnixi, wheren∑i=1

cni = 0,

n∑i=1

cnixi 6= 0.

Moreover two theorems concerning asymptotic normality of α∗n werestated and proven.

The talk is based on a joint work [1] with Alexander I. Sakhanenko(Novosibirsk State University and Sobolev Institute of Mathematics,Novosibirsk).

[1] E. N. Savinkina, A. I. Sakhanenko. Explicit estimators of an unknownparameter in a power regression problem.Siberian Electronic Mathematical Reports, 11 (2014), 725-733.

5This work is supported by the RFBR-grant #15-01-07460a.


On the real accuracy of approximations inCLT

Senatov V.V.

Let X1, X2, . . . � independent random variables with zero mean, unitvariance and common distribution P . At su�ciently wide conditions existdensity pn(x) of distribution of (X1 +· · ·+Xn)/

√n, pn(x)→ ϕ(x), n→∞,

where ϕ(x) = e−x2/2/√

2π, and Bk,n = 12π

∫∞−∞ |t|

kµn−1(t/√n) dt →

12π

∫∞−∞ |t|

ke−t2/2 dt, n → ∞, where µ(t) = max{|f(t)|; e−t2/2}, f(t)

� characteristic function of P . We denote Hk(x) = (−1)kϕ(k)(x)/ϕ(x)� Chebyshev�Hermite polynomials, αk =

∫∞−∞ xk P (dx), θk =∫∞

−∞Hk(x)P (dx), k = 0, 1, . . .. We need only the numbers θ4 = α4 − 3and θ6 = α6 − 15α4 + 30, as well as the numbers ||θ4|| = α4 + 3 and||θ6|| = α6 + 15α4 + 60.

We discuss new estimates of the accuracy of approximations in thelocal form of CLT for densities of symmetric distributions. In constructingof these estimates were used asymptotic expansions.

The following inequalities hold

|pn(x)− ϕ(x)| 6 ||θ4||4!n

B4,n =α4 + 3

4!nB4,n,∣∣∣∣pn(x)− ϕ(x)

(1− 3

4!nH4(x)

)∣∣∣∣ 6 α4

4!nB4,n +

15

6!n2B6,n ∼

α4

8√

2πn, n→∞,

(1)

and ∣∣∣∣pn(x)− ϕ(x)(1 +

θ4

4!nH4(x)

)∣∣∣∣ 6 ||θ6||6!n2

B6,n +1

2

|θ4|4!n

||θ4||4!n

B8,n,∣∣pn(x)− ϕ(x)∣∣ 6 |θ4|

4!n

∣∣H4(x)∣∣ϕ(x) +

||θ6||6!n2

B6,n +1

2

|θ4|4!n

||θ4||4!n

B8,n. (2)

These results are compared with the real accuracy of approximationsfor the uniform on [−

√3;√

3] distribution for the densities pn(x) of which isknown explicit formula. The inequality (1) not be improved in some sensebut for uniform distribution the approximation the density pn(x) using


function ϕ(x)(1− 1

8nH4(x))is worse approximation using ϕ(x). In (2) for

wide class of distributions P the �rst term in the right side is equivalentfor n→∞ to its left side.

Analytic di�usion processes: de�nition,properties, limit theorems

N.V.SmorodinaPDMI RAS

St.-Petersburg State University,St.-Petersburg, Russia

We introduce a concept of an analytic di�usion process. We de�ne sucha process as a limit of a sequence of random walks but we understandthis limit not in the sense of convergence of measures but in the sense ofconvergence of generalized functions. In terms of analytic di�usion processeswe construct a probabilistic approximation of the Cauchy problem solutionsfor Schr�odinger type evolution equations having in the right hand sideelliptic operators with variable coe�cients.


On approximation in Koul's theorem forweighted empirical processes 6

Olga A. SukhovershinaNovosibirsk State University, RussiaEmail: [email protected]

Let, for each n = 1, 2..., we have independent random variablesηn1, ..., ηnn with distribution functions Gni(x) = P(ηni 6 x). We considerweighted empirical distribution functions

Fn(t) :=∑

i6ndni{I(ηni 6 t)−Gni(t)}, t ∈ [0, 1]

where dn1, ..., dnn are some non-random weights, satisfying the conditions∑i6n

d2ni = 1, µn = max

i6n|dni| → 0, τn = sup

t

∑i6n

d2niP(ηni = t)→ 0.

Theorem 4. For any integers n ≥ 1, m ≤ 1/τn and real ε > 0 onthe same probability space with empirical process Fn, we can construct anaccompanying Gaussian process Wn such that

P(||Fn −Wn|| > C1ε

)6m3/2µnε3

+ 26m exp{− ε2m

1 + εmµn

}.

This estimate also allows us to obtain Koul's theorem [1, p.16] undermore general assumptions.

The talk is based on a joint work [2] with A.I. Sakhanenko.

[1] Hira L. Koul. Weighted Empirical Processes in Dynamic NonlinearModels, Springer-Verlag, New York, 2002.

[2] Sakhanenko A.I., Sukhovershina O.A., On accuracy of approximationin Koulï¿½s theorem for weighted empirical processes. SiberianElectronic Mathematical Reports, 12 (2015), 784-794.



Inequalities for the sojourn time of randomwalk above a certain boundary

Tarasenko A.S.Sobolev Institute of MathematicsNovosibirsk State University

Under general conditions, we obtain inequalities for moments of sojourntime of a random walk over linear boundary. We �nd asymptotics of thesemoments for random walks with regular or semi-exponential distributionof summands.

Mass-Stationarity, Shift-Coupling, andBrownian Motion

Hermann ThorissonUniversity of Iceland

Palm versions w.r.t. stationary random measures are mass-stationary,that is, the origin is at a typical location in the mass of the random measure.For a simple example, consider the stationary Poisson process on the lineconditioned on having a point at the origin. The origin is then at a typicalpoint (at a typical location in the mass) because shifting the origin to thenth point on the right (or on the left) does not alter the fact that the inter-point distances are i.i.d. exponential. Another (less obvious) example is thelocal time at zero of a two-sided standard Brownian motion.

In this talk we shall consider mass-stationarity on the line and the shift-coupling problem of how to shift the origin from a typical location in themass of one random measure to a typical location in the mass of anotherrandom measure. Applications include unbiased Skorohod embedding andembedding the Brownian bridge into the Brownian motion. In his talk,Guenter Last will focus on applying this theory to �nding an extraexcursion. The present talk is based on joint work with him, Peter Moertersand Wenpin Tang.


[1] Last, G. and Thorisson, H. (2009). Invariant transports of stationaryrandom measures and mass-stationarity. Annals of Probability 37,790�813.

[2] Last, G., Moerters, P. and Thorisson, H. (2014). Unbiased shifts ofBrownian motion. Annals of Probability 42, 431-463.

[3] Pitman, J. and Tang, W. (2015). The Slepian zero set, and Brownianbridge embedded in Brownian motion by a spacetime shift. ElectronicJournal of Probability 61, 1-28.

[4] Last, G., Tang, W. and Thorisson, H. (2016). Transporting randommeasures on the line and embedding excursions into Brownianmotion. In preparation.

Properties of renewal matrices based ondistribution functions with regularly varying

tails

Valentin Topchii 7

Sobolev's Institute of Mathematics of SB RAS (Omsk Branch)

Let a matrix M :=(mij

)ni,j=1

with non-negative entries beindecomposable and aperiodic, with the largest eigenvalue equal to 1.

Fix a family of distribution functions Gi(t) such that for some 0 < n0 <n and all i ∈ I0 := 1, 2, · · · , n0 and j ∈ I1 := n0 + 1, · · · , n

1−Gi(t) := qi(t) = o(t−2), 1−Gj(t) := qj(t) = t−βj `j(t), βj ∈ (0, 1],

where `j(t) is a function slowly varying at in�nity. Suppose that qj(t) =O(qn(t)), for all j > n0.

7Supported by program FSI �Development of the methods for investigating somestochastic models aimed towards population and biomedical applications�


De�ne the main matrix and the renewal matrix for one as follows

M(t) :=(mijGi(t)

)ni,j=1

, U(t) :=

∞∑k=0

M∗k(t),

were M∗k(t) is the k-th convolution of matrix M(t).We proposed the family of conditions su�cient for regular varying at

the in�nity of increments of the �rst and of the second orders for U(t).In view ofU(t) ∼ cβn

∫ t0qn(u)duD, were cβn

is a some constant dependsof βn and D is a some matrix that is de�ned with the help of M, the mainpart of increments are de�ned in terms of ones for

∫ t0qn(u)du.

All results are based on Banach-algebraic representations of complex-valued measures and V.A. Shurenkov's representation of multi-dimensionalrenewal equation through one-dimensional ones (Theory Probab. Appl.,1975). More precisely, we deal with a class real-valued measures that maybe represented as di�erences of positive measures. The method has beenoriginally developed by B.A. Rogozin and M.S. Sgibnev (Siberian Math.Journal, 1979, 1988).


Construction and investigation of statisticalestimators for pressure-derivatives8

Evgeniya TsebukhovskayaNovosibirsk State University, RussiaEmail: [email protected]

Suppose we are given 2n+ 1 independent observations

Yk = f(t0 + khn) + εk, k = 0,±1,±2, . . . ,±n,where f is unknown function. The subject of the research is a problem ofevaluation of pressure-derivative l(t0), which arises during measurementsof pressure f(·) in a case of hydrodynamic well testing. Notion of pressure-derivative was introduced in a paper [1]: l(t) := d log f(t)

d log t = t f′(t)f(t) . If we

suppose that f = ctp then estimator for pressure-derivative is also estimatorfor unknown parameter p, which is important in well testing.

To �nd estimators for pressure-derivatives we use statistics:

f̂n(t0) =

∑nk=−n uk,nYk∑nk=−n uk,n

and f̂ ′n(t0) =

∑nk=1 vk,n

Yk−Y−k

2khn∑nk=1 vk,n

.

From mathematical point of view investigated problem consists inoptimization of some functionals, responsible for accuracy sought-forevaluation, and on search of constants {uk,n} and {vk,n}, which giveminimum to these functionals. In the work conditions on constants {uk,n}and {vk,n} are found under which estimator l̂(t0) = t0f̂

′n(t0)/f̂n(t0) is

consistent and asymptotically optimal in some sense.The talk is based on a joint work [2] with A.I. Sakhanenko.

[1] Hosseinpour-Zonoozi N., Ilk D., Blasingame T.A. The PressureDerivative Revisited � Improved Formulations and Applications //paper SPE 103204 presented at the 2006 SPE Annual TechnicalConference and Exhibition, San Antonio, Texas, USA., 24-27September 2006. Proceedings, 7, pp. 4277-4303.

[2] Sakhanenko A.I., Tsebukhovskaya E. Construction and investigationof statistical estimates of pressure-derivative function // Submittedto Journal of Applied and Industrial Mathematics



On non-asymptotic results for MANOVA teststatistics with high-dimensional data9

Ulyanov V.V.Lomonosov Moscow State University


Lipatiev A.A.

We consider famous MANOVA test statistics: the likelihood ratio teststatistics TLR, Lawley-Hotelling's generalized statistics TLH , and Bartlett-Nanda-Pillai's test statistics TBNP when a number n of observations iscomparable with its dimension p, e.g. p/n → c ∈ (0, 1). Since the exactdistributions of these statistics are complicated and not easy to treat, weneed some approximations for them.

The errors of approximations could be described either in asymptoticway as an order of a remainder term with respect to n and p (see, e.g., [1])or in non-asymptotic form as a bound for remainder term with explicitdependence on n, p and moment characteristics of random elements orobservations (see, e.g., Chapters 13-16 in [2]).

The non-asymptotic results for TLR were obtained in [3]. The talk isdevoted to similar non-asymptotic bounds for distributions of TLH andTBNP .

[1] Wakaki H., Fujikoshi Y., Ulyanov V. V. Asymptotic expansions ofthe distributions of MANOVA test statistics when the dimension islarge // Hiroshima Math. J., 2014. Vol. 44. No. 3. P. 247�259.

[2] Fujikoshi Y., Ulyanov V. V., Shimizu R.Multivariate statistics. Highdimensional and large-sample approximations. � Hoboken, N.J.: JohnWiley and Sons, 2010. 533 p.

[3] Ulyanov V. V., Wakaki H., Fujikoshi Y. Berryâ��Esseen boundfor high dimensional asymptotic approximation of Wilks' Lambdadistribution // Statistics & Probability Letters, 2006. Vol. 76. P.1191�1200.

9The research was supported by RFBR grant, No. 14-01-00500, and by RSCF 14-11-00364.


Moments for multidimensional criticalBellman-Harris processes where particles

life-lengths have in�nite mean and regularlyvarying tails of di�erent order

Valentin Topchii 10

Sobolev's Institute of Mathematics of SB RAS (Omsk Branch)

Vladimir Vatutin 11

Steklov's Mathematical Institute of RAS

The n-types Bellman�Harris branching process Z(t) =(Z1(t), Z2(t), · · · , Zn(t)

)may be described as follows. A particle of

type i ∈ {1, 2, · · · , n} has life-length distribution Gi(t), and at the end ofits life it produces ξij particles of all j ∈ {1, 2, · · · , n} types particles withthe moments mij = Eξij and �nite second moments Eξijξis < ∞ for alli, j, s ∈ {1, 2, · · · , n}. Each particle ever born behaves in a similar manner,and it lives and produces o�spring independently of the co-existingparticles and the past history of the process. We assume that Z(t) isindecomposable, aperiodic and critical. Suppose that for some 0 < n0 < nand all i ∈ I0 := 1, 2, · · · , n0 and j ∈ I1 := n0 + 1, · · · , n

1−Gi(t) := qi(t) = o(t−2), 1−Gj(t) := qj(t) = t−βj `j(t), βj ∈ (0, 1],

where `j(t) is a function slowly varying at in�nity. Suppose that qj(t) =O(qn(t)) for all j ∈ I1 and Zi(t) is the process that stats with a singleparticle of type i. Set ∆G(t) := G(t+ ∆)−G(t) for a �xed ∆ > 0 and letµs(t) =

∫ t0qs(u)du. Note that, as t → ∞, µs(t) ∼ µs < ∞ for s ∈ I0, and

µs(t)→∞ is a regularly varying at in�nity function with index 1− βs, fors ∈ I1.

If, in particular, βn = 1, then we assume in addition that ∆Gi(t) =o(t−2 min{1, `n(t)}

), and if βn ∈ (0, 0.5], the we assume that ∆Gi(t) ≤

Ct−1−βn`n(t), for some C > 0 and all i ∈ I1. Then the following resultholds:

EZij(t) ∼ cijµj(t)µ−1n (t), as t→∞,

10Supported by grant RFBR 14-01-00318 and program of RAS �Modern methodsapproximability of models, algorithms and theories�

11Supported by program of RAS �Mathematical problems of modern control theory�


where cij are well-de�ned constants.Under further assumptions, we provide the asymptotics for E(Zij(t))

2

and for the increments of EZij(t).

Of averaged control for 1D ergodic di�usion

S. V. AnulovaInstitute of Control Sciences, RAS, Moscow, Russia


H. MaiCREST, ENSAE ParisTech, Paris, France


A. Yu. Veretennikov12

Presenter and corresponding author: University of Leeds, UK; & HigherSchool of Economics, & Institute for Information Transmission Problems,

Moscow, RussiaEmail: [email protected]

Consider a 1D controlled non-degenerate di�usion

dXαt = b(α(Xα

t ), Xαt ) dt+ σ(α(Xα

t ), Xαt ) dWt, t ≥ 0, Xα

0 = x.(1)

The range of control values {u} ≡ U ⊂ R is a compact set, the set of allMarkov strategies α(·) is denoted by K. For every α ∈ K the solution Xα

is assumed to be ergodic uniformly for all strategies (but non-uniformlywith respect to initial conditions). Given a Borel bounded (can be relaxed)function f = f(α, x), α ∈ U, x ∈ R, we want to to minimise the averagedcost function,

ρ(x) := infα∈K

ρα(x) ≡ infα∈K

limT→∞

1

T

∫ T

0

Exf(α(Xαt ), Xα

t ) dt. (2)

12The work for the third author was prepared within the framework of a subsidygranted to the HSE by the Government of the Russian Federation for the implementationof the Global Competitiveness Program.


In fact, because of the �good ergodicity� (conditions to be presented), ρα(x)does not depend on x and is a constant ρα, and so is ρ. It is proved that thecost value ρ is the component of the pair (V, ρ), which is a unique solutionin an appropriate Sobolev sense of the ergodic HJB or Bellman's equation,

infu∈U

[LuV (x) + fu(x)− ρ] = 0, x ∈ R, (3)

where Lu is the extended generator corresponding to the equation (1) withα(x) ≡ u. Here V is unique up to an additive constant, while ρ is unique inthe standard sense. It is also shown that a �reward improvement algorithm�(RIA) � the details to be presented in the talk � provides a sequence ofconvergent costs, ρn → ρ, n→∞. To the best of the authors' knowledge,earlier works � see [1, 2] � did not involve a di�usion coe�cient with control.

[1] A. Arapostathis, V. S. Borkar, A relative value iteration algorithm fornondegenerate controlled di�usions. SIAM J. Control Optim. 2012,50(4), 1886-1902.

[2] A. Arapostathis, On the policy iteration algorithm for non-degeneratecontrolled di�usions under the ergodic criterion, in: Optimization,Control, and Applications of Stochastic Systems, D. Hern�andez & A.Minj�arez-Sosa Eds., Springer, New York, 2012, 1-12.


Controlled continuous time random walks andtheir extensions

Maria Veretennikova

For the �rst time we have studied Controlled Continuous Time RandomWalks. Extending the existing optimal control theory formalism we proveconvergence of payo� functions of the scaled processes to the solution ofan optimization problem and a fractional in time Hamilton Jacobi Bellmanequation. We present the well-posedness analysis for the fractional Cauchyproblem. We also extend the theory to a more broad spectrum of processes.

First passage times for random walks withnon-identically distributed increments

Vitali WachtelAugsburg, Germany

We consider random walks with independent but not necessarilyidentical distributed increments that belong to the domain of attractionof the normal distribution. We will discuss the asymptotic behaviour of�rst-passage times of such random walks. (This is a joint work with D.Denisov and A. Sakhanenko.)


Arak Inequalities for Concentration Functionsand the Littlewood�O�ord Problem

Zaitsev A.Yu.St. Petersburg Department of Steklov Mathematical Institute


We discuss the behavior of concentration functions of weighted sums ofindependent random variables with respect to the arithmetic structure ofcoe�cients. Recently, Tao and Vu [4] and Nguyen and [3] formulated a so-called Inverse Principle in the Littlewood�O�ord problem. We discuss therelations between this Inverse Principle and a similar principle formulatedfor sums of arbitrarily distributed independent random variables formulatedby T. Arak in the 1980's (see [1] and [2]).

[1] T. V. Arak, A. Yu. Zaitsev, Uniform limit theorems for sums ofindependent random variables, Trudy MIAN, 174 (1986) (in Russian),English translation in Proc. Steklov Inst. Math., 174 (1988).

[2] Yu. S. Eliseeva, F. G�otze, A. Yu. Zaitsev, Arak Inequalitiesfor Concentration Functions and the Littlewood�O�ord Problem.arXiv:1506.09034 (2015).

[3] Hoi Nguyen, Van Vu, Optimal inverse Littlewood�O�ord theorems.Adv. Math. 226 (2011), 5298�5319.

[4] T. Tao, Van Vu, Inverse Littlewood�O�ord theorems and thecondition number of random discrete matrices, Ann. of Math. (2),169 (2009), 595�632.


Convex hulls of random walks, hyperplanearrangements, and Weyl chambers

Dmitry ZaporozhetsS.-Petersburg

We give an explicit formula for the probability that the convex hullof an n-step random walk in Rd with centrally symmetric density ofincrements does not contain the origin. This is a distribution-free resultthat extends the one-dimensional formula of Sparre Andersen (1949):any symmetric continuously distributed random walk stays positive with

probability(2n− 1)!!

2nn!.

This probabilistic problem is shown to be equivalent to the followinggeometric one: Find the number of Weyl chambers of type Bn intersectedby a generic linear subspace in Rn of codimension d. We solve this geometricproblem using the theory of hyperplane arrangements.

Based on joint work with Zakhar Kabluchko and Vlad Vysotsky:http://arxiv.org/abs/1510.04073


Segment recombinations and random sharingmodels

Sergei ZuyevUniversity of Technology and University of Gothenburg, Department of

Mathematical Sciences, Gothenburg, SwedenEmail: [email protected]

Consider a renewal point process on the line and divide each of thesegments it de�nes in proportion given by i.i.d. realisations of a �xeddistribution supported by [0, 1]. Now recombine the obtained pieces of thesegments by joining the neighbouring ones, so that the division points arenow the separation points between the new segments. We ask ourselves forwhich renewal processes and which division distributions the division pointsfollow the same renewal process distribution? An evident case is that ofequal length segments and a degenerate division distribution. Interestingly,the only other possible case is when the increments of the renewal process isGamma and division points are Beta-distributed. In particular, the divisionpoints of a Poisson process is again Poisson, if the dividing distribution isBeta(r, 1− r) for any 0 < r < 1.

We show that a similar situation arises in the random sharing modelwhen a countable number of `cites' exchange randomly distributed parts oftheir `wealth' with neighbours. More generally, Dirichlet distribution arisesin these models as the distribution leading to a �xed point. We also showthat the �xed points of the random sharing are attractors meaning thatstarting with a non-equilibrium con�guration distribution will converge tothe equilibrium.

A joint work with Anton Muratov.


On the coupling method for non-discreterenewal and regenerative processes

G. A. Zverkina

A new modi�cation of the coupling method is proposed for regenerativeprocesses under assumptions of existence of the density for the distributionof the regeneration period length and of its certain positiveness. Thismodi�cation allows to obtain computable bounds for the rate ofconvergence of the backward renewal time distributions and eventually ofthe whole process to the corresponding stationary regimes. As examples,processes with exponential and polynomial moments of the regenerationperiod distributions are considered.


List of Participants

Afanasyev, Valeriy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Igor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Alexandr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Konstantin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Onno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Mikhail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Denis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Shakir K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Sergey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Carl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected] der Hofstad, Remco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Ildar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Anzhelika . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Haya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], O�er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29o�[email protected]


Konstantopoulos, Takis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Dmitry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], G�unter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Mikhail A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Yuliana Yu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Nelly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Vladimir I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Vyacheslav S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Alexey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Thomas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Masakiyo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Anatolii A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected]�orters, Peter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Vyacheslav E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Thomas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42thomas.mountford@ep�.chNasyrov, Farit S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Guodong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected]


Presman, Ernst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Kavita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Maria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Philippe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Alexander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Alexander I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Ekaterina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Vladimir V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Natalia V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Olga A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Anton S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Hermann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Valentin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Evgeniya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Vladimir V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Alexandr Yu. . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Maria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected]


Wachtel, Vitali . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Adrei Yu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Dmitry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Sergei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected], Galina A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected]

vi international conference - nsc.rumath.nsc.ru/lbrt/v1/conf2016/program.pdf · 2016. 8. 11. ·...

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