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Introduction to Stochastic-Process LimitsProbability Seminar
Yunan LiuIndustrial and Systems Engineering
North Carolina State University
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Why Do We Care?
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Motivation
• Simpleapproximationsfor complicatedstochastic processes
• Explainstatistical regularity (appropriate scaling)
• Engineeringperspective: applications to service systems
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Illustration
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Unit Poisson Process
N(t) is a Poisson (counting) process with rate 1
PlotN(t) as a function of timet in [0, T ]
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0 2 4 6 8 100
1
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8
9
10
t
N(t
)
N(t) in [ 0 , T ] for T = 10
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0 20 40 60 80 1000
10
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100
t
N(t
)
N(t) in [ 0 , T ] for T = 100
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0 2 4 6 8 10
x 104
0
1
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10x 10
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t
N(t
)
N(t) in [0, T] for T = 105
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What’s Going on?
The plotter scales bothtimeandspace.
What we see is a sample path of:
N(t) = 1nN(nt) wheren = 104.
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More Plots
PlotN(t)− t as a function of timet in [0, T ]
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0 2 4 6 8 10
−3
−2
−1
0
1
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t
N(t
) −
t
N(t) − t in [ 0, T ] for T = 10
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0 20 40 60 80 100−15
−10
−5
0
5
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t
N(t
)
N(t) − t in [ 0 , T ] for T = 100
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0 2 4 6 8 10
x 104
−300
−200
−100
0
100
200
300
t
N(t
) −
t
N(t) − t in [ 0, T ] for T = 10 5
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What’s Going on?
The plotter scales bothtimeandspace.
What we see is a sample path of:
N(t) = 1√n(N(nt)− nt) wheren = 104.
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A Little Measure Theory
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Measurable Space
B(S) is aσ-algebraof subsets ofS if:
1.S ∈ B(S)
2. If F ∈ B(S) thenF c ∈ B(S)
3. If F1, F2, F3, ... ∈ B(S), then∪iFi ∈ B(S)
(S, B(S)) is a measurable space
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Measure
• µ : (S, B(S)) → [0,∞] is ameasureif µ(φ) = 0 and :
µ(∪∞i=1Fi) = Σ∞i=1µ(Fi) ,
where(Fn : n ≥ 1) is a sequence of pairwise disjointsets inB(S).
• µ is aprobability measureif µ(S) = 1.
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Measurable Function
h : (S1, B(S1)) → (S2, B(S2)) is measurableif:
h−1(A) = s ∈ S1 : h(s) ∈ A ∈ B(S1),
for A ∈ B(S2).
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Random Element
X is measurable
• If S ≡ R, thenX is arandom variable
• If S is a function space (e.g.,D), thenX is astochasticprocess
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Stochastic Process Limits
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Almost Sure Convergence
X1, X2, X3, ... is a sequence of stochastic processes (e.g., inC).
Xn → X w.p.1⇔ P (‖Xn −X‖T → 0) = 1,∀ T ≥ 0,
where,
||Xn −X||T = sup0≤t≤T |Xn(t)−X(t)| .
We say thatXn converges toX a.s. u.o.c.
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Weak Convergence of Probability Measures(S, m) metric space
Pn : n ≥ 1 on (S, m) converges weaklyto P if:
limn→∞
∫S
fdPn =
∫S
fdP, ∀f ∈ C(S).
And we write,
Pn ⇒ P .
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Prohorov Topology
P (S) is the space of probability measures onS.
We can define a metricπ onP (S), such that:
π(Pn, P ) → 0 in P (S) ⇔ Pn ⇒ P, .
π is theProhorov metriconP (S).
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Probability LawLet X : (Ω, F, P ) → (S, B(S)) be a random element.
Theprobability lawof X is defined as:
PX−1(A) ≡ P (X−1(A)) ≡ P (w ∈ Ω : X(w) ∈ A),
for A ∈ B(S).
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Weak Convergence of Random ElementsLet Xn, X : (Ω, F, P ) → (S, B(S)) be random elements.
We say thatXn ⇒ X if
PX−1n ⇒ PX−1
m
E[f (Xn)] → E[f (X)],∀f ∈ C(S).
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A Useful Theorem
Skorohod Representation Theorem
If Xn ⇒ X in (S, m), then there existsXn andX, defined on a common probabilityspacesuch that:
Xn ∼ Xn andX ∼ X ,
and
Xn → X w.p.1.
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Proving Limits?
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Continuous Mapping Approach
Simple Continuous-Mapping theorem (CMT)
If Xn ⇒ X in (S, m) andg : (S, m) → (S′, m′)is continuous, then:
g(Xn) ⇒ g(X) in (S′, m′) .
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Generalized Continuous Mapping Theorem
Let gn : (S, m) → (S ′, m′) bemeasurablefunctions.
Supposegn(xn) → g(x) wheneverxn → x for x ∈ E ⊆ S such thatP (X ∈ E) = 1
Xn, X random elements in(S, m).
If Xn ⇒ X thengn(Xn) ⇒ g(X)
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Proof of Generalized Continuous Mapping TheoremTheorem. Let gn : (S, m) → (S′, m′) bemeasurablefunctions.
Supposegn(xn) → g(x) wheneverxn → x for x ∈ E ⊆ S such thatP (X ∈ E) = 1
If Xn ⇒ X thengn(Xn) ⇒ g(X).
Proof.(Skorohod) ⇒∃Xn ∼ Xn, X ∼ X defined on the same spacesuch that:
Xn → X w.p.1,
whereP (X ∈ E) = 1.
Deterministic Framework: gn(Xn) → g(X) w.p.1
w.p.1 convergence impliesweakconvergence:gn(Xn) ⇒ g(X)
By equality of distributions,
gn(Xn) ⇒ g(X).
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Another Useful TheoremDonsker’s Theorem
• Xn : n ≥ 1 IID random variables
• m = E[X1] andV arX1 = σ2 < ∞
Define,
Sn(t) =1√n
(
bntc∑i=1
Xi −mnt), t ≥ 0 .
Then,
Sn ⇒ σB ,
whereB denotes standard Brownian motion.
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Example: Renewal ProcessAsymptotics
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A Fundamental Process
Define,
X(t) =
btc∑i=1
ξi, t ≥ 0 ,
whereξi, i ≥ 1 sequence of IID nonnegative random variables.
And the inverse process
Y (t) = sups ≥ 0 : X(s) ≤ t .
Then,Y (t) is a renewal counting process!
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Functional SLLN for Renewal Processes
Standard SLLN for X⇓
Xn(t) ≡ 1
nX(nt) → X(t) ≡ mt w.p.1 u.o.c ,
⇓ (CMT)
Y n(t) ≡ 1
nY (nt) → Y (t) ≡ t/m w.p.1 u.o.c ,
wherem = E[ξi] > 0 (finite).
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Functional CLT for Renewal ProcessesIf,
Xn(t) =√
n(Xn(t)− X(t)) , t ≥ 0,
Y n(t) =√
n(Y n(t)− Y (t)) , t ≥ 0,
whereX(t) = mt andY (t) = t/m.
Then,
Xn(t) ⇒ X(t) ≡ σB(t)
Y n(t) ⇒ − 1
mX(t/m).
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Random-Time Change Theorem
Random-Time Change Theorem
Let Un : n ≥ 1 andVn : n ≥ 1 be two sequences of stochastic processes inD.
If Vn is nondecreasing,Vn ⇒ v (deterministic), andUn ⇒ U , then,
Un(Vn) ⇒ U(v) ,where,
Un(Vn) ≡ Un(Vn(t)), t ≥ 0 ,
and,
U(v) ≡ U(v(t)), t ≥ 0 .
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Proof of Functional CLTTheorem. Xn(t) =
√n(Xn(t)− X(t)) andY n(t) =
√n(Y n(t)− Y (t))
Then,
Xn(t) ⇒ X(t) ≡ σB(t) andY n(t) ⇒ − 1mX(t/m)
Proof.
Donsker’s: Xn ⇒ X
FSLLN: Xn(t) → X(t) a.s. u.o.c. andY n(t) → Y (t) a.s. u.o.c
Random-Time Change: Xn(Y n) =√
n[Xn(Y n)− X(Y n)] ⇒ X(Y )
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Proof of Functional CLT (Cont’)Theorem. Xn(t) =
√n(Xn(t)− X(t)) andY n(t) =
√n(Y n(t)− Y (t))
Then,
Xn(t) ⇒ X(t) ≡ σB(t) andY n(t) ⇒ − 1mX(t/m)
Proof (Cont’).
Y n =√
n(Y n − Y )
=
√n
m[X(Y n)− X(Y )]
=
√n
m[X(Y n)− Xn(Y n) + Xn(Y n)− X(Y )]
=
√n
m[X(Y n)− Xn(Y n)] +
√n
m[Xn(Y n)− X(Y )]
⇒ − 1
mX(Y ) ,
since we can show that√
n[Xn(Y n)− X(Y )] → 0 a.s. u.o.c.
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Thanks!