moment problem and density questions akio arimoto
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Moment Problem and Density Questions Akio Arimoto. Mini-Workshop on Applied Analysis and Applied Probability March 24-25,2010 at National Taiwan University. March 24-25,2010 at N T U. Stationary Stochastic Process PredictionTheory Truncated Moment Problem Infinite Moment Problem - PowerPoint PPT PresentationTRANSCRIPT
Moment Problem and Density Questions
Akio Arimoto Mini-Workshop on Applied Analysis and Applied Probability
March 24-25,2010
at National Taiwan University
March 24-25,2010 at N T U
Stationary Stochastic Process PredictionTheory Truncated Moment Problem Infinite Moment Problem Polynomial Dense N-extreme Measure Conclusion
Topics ,Key words
Stationary Stochastic Sequences
Let
, 0, 1, 2,nX n
, ,F P Probability space
Random variables with time variable n
0,n nEX X dP
,n m n mPX X EX X n m
2
0
ikk e d
Spectral representation
Positive Borel Measure
weakly stationary
March 24-25,2010 at N T U
Discrete Time Case( Time Series)
Stationary stochastic process
, : , ,X t t
, , 0EX t X t P d
, , ,
Pt s EX t X s X t X s
i tt e d
Spectral representation
(Bochner’s theorem)
March 24-25,2010 at N T U
Continuous Time Case
Conditions of deterministic
2
0
logw d
2
log
1
wd
sd w d d
March 24-25,2010 at N T U
Conformal mapping from the unit circle to upper half plane
nX is deterministic
X t is deterministic
Transform the probability space into the function space
2
2
,0
, ,i n m n mn m P L T
X X n m e d z z
0 0 1 1 0 1... ... nn n na X a X a X a a z a z
2
0
, , 0,1, 2, ,ik ik kkX e Z e z k n
, ,F P 2 ,L T
March 24-25,2010 at N T U
Discrete time case
Space of random variables
with finite variance
Space of square
summable functions
0 0 1 1 0 1... ... nn n na X a X a X a a z a z
Y f z
2
22
0 0 1 1 0 1
0
... ... nn n nE Y a X a X a X f z a a z a z d
isometry isometry
20 0 1 1 0 1 ,
... ... nn n nP L T
Y a X a X a X f z a a z a z
Statistical Estimation error = Approximation error
March 24-25,2010 at N T U
Discrete time case
Kolmogorov-Szego’s Theorem of Prediction
1 2
22
1, ,0
inf 1 exp loga a
a z w d w d
Kolmogorov’s Theorem
Szegö’s Theorem:(Kolmogorov refound)
,sd w d d :d Lebesgue measure
1 2 1 2
2 22 2
1 1, , , ,
0 0
inf 1 inf 1a a a a
a z d a z w d
March 24-25,2010 at N T U
Discrete time
Prediction Error
2
2
1 1 2 2
0
inf exp logk
m m ma
E X a X a X w d
2
0
0, logif w d
2 2
0 0
exp log , logw d if w d
March 24-25,2010 at N T U
deterministic
indeterministic
History
A.N.Kolmogorov , Interpolation and Extrapolation of Stationary Sequences, Izvestiya AN SSSR (seriya matematicheskaya),5 (1941), 3-14
(Wiener also had obtained the same results independently during the World War II and published later the following )
N. Wiener, Extrapolation, Interpolation, and Smoothing of Statioanry Time Series, MIT Technology Press (1950)
Kolmogorov Hilbert Space (astract Math.)
Wiener Fourier Analysis (Engineering sense)
March 24-25,2010 at N T U
Szegö’s Alternative
Either
w d Absolute continuous part of d
2
log
1
wd
and
2 0
0
T
T
L Z Z
where
2, ,ab i tZ linearspanof e a t b in L
indeterministic
March 24-25,2010 at N T U
Continuous time
or else
2
log
1
wd
2 0
0
T
T
L Z Z
Deterministic case
then
Continuous time
2, ,ab i tZ linearspanof e a t b in L
March 24-25,2010 at N T U
We can have an exact prediction from the past
This book deals with the relation between the past and future of stationary gaussian process, Kolmogorov and Wiener showed ・・・The more difficult problem, when only a finite segment of past known, was solved by Krein....spectral theory of weighted string by Krein and Hilbert space of entire function by L. de Branges…Academic Press,1976Dover edition,2008
March 24-25,2010 at N T U
Problem of Krein
, , 2 0,X t T t
Predict the future value , , 0X t t
i t Te
on T i tZ span of e t T
Finite Prediction
From finite segment of past
Compute the projection of
Krein’s idea=Analyze String and spectral function
March 24-25,2010 at N T U
Moment Problem Technique ( see Dym- Mckean book in detail)
2
0
,ik k
T
k e d z d
0 , 1 , 2 ,
Moment Problem
0 , 1 , 2 , N
uniquely determined
March 24-25,2010 at N T U
indeterminated
iT z e
Representing measure
2
0
ikk e d
0 , 1 , 2 , N is called the representing measure of
if
We particularly have an interest to find
the extreme points of
March 24-25,2010 at N T U
2
0
0 , 1 , , , 0,1,2,ikM N k e d k N
a set of representation measures( convex set)
0 , 1 , ,M N
Truncated Moment Problem
March 24-25,2010 at N T U
0 0
0,N N
j kj k
j k a a
2
0
0N
jj
a
0 1, , , Na a afor any such taht
0 , 1 , 2 , N
Positive definite
Find representing measures of which moments are
And characterize the totality of representation measures
0 , 1 , 2 , N
Properties of Extreme Points
0 , 1 , ,M N is an ex t reme point of conves set
1 { 0, 1, 2, , }k iL d linear span z k N z e
is the representing measure for a singular extension of
0 , 1 , 2 , N
March 24-25,2010 at N T U
Polynomial dense in 1 2L d L d
Singularly positive definite sequence Arimoto,Akio; Ito, Takashi,
Singularly Positive Definite Sequences and Parametrization of Extreme Points. Linear Algebra Appl. 239, 127-149(1996).
March 24-25,2010 at N T U
Trucated Moment Problem
Singular positive definite sequence 0 1 1, , , ,M Mc c c c
0 1, , , Mc c c is positive definite
0 1 1, , , ,M Mc c c c is nonegative definite but positive definite
Is singular positive definite
March 24-25,2010 at N T U
Theorem: extreme measures is an extreme point of 0 1, , , NM c c c
2
0
,ikkd e d
0,1,2 1k M
0 1 1, , , ,M Md d d d is singular extenstion of
0 1, , , Nc c c 2N M N
( . . ,0 )k ki e d c k N
March 24-25,2010 at N T U
Extreme points of representing measures Let
0
N
M k kk
E z P P z
Singularly Positive Sequence
determines uniquely measure as 1
21
1k
k
N
aa
kNE
where , 1, 2, 1ka k N are zeros of a polynomial 1NP z
March 24-25,2010 at N T U
simple roots on the unit circle . , 1, 2, 1ka k N 1ka
0 1, , NP z P z P z
Orthonormal polynomials
2
0
, i n mn mz z e d
0 1, , , Nc c c
Hamburger Moment Problem
(*) , 0,1,2...,kks x d x k
, 0,1,2,ks k Find satisfying (*)
ks is a moment sequence of
March 24-25,2010 at N T U
Infinite Moment Problem
where has infinite support
Achiezer : Classical Moment Problem
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Riesz’s criterion
R z
2sup 1L
p PR z p z p
0R z
(1’)
(1)
March 24-25,2010 at N T U
For some
For any \ ,z
0 \ ,z
The Logarithmic Integral
(2)
2
log
1
R xdx
x
This is a common formula which appears in the moment problem and the prediction theory.
March 24-25,2010 at N T U
( 4 ) is dense in P 2L 21d x x d x
(5)
is dense in
iP x i p p P
2L
March 24-25,2010 at N T U
Is determinate(3)
(1) (2) (3) (4) (5) are equivalent
Equivalence
March 24-25,2010 at N T U
has been proved by Riesz, Pollard and Achiezer
Important Inequality
2
11 1 1 1inf
1 Imp PL
zp x
z R z x z z R z
21d x x d x
P polynomials
March 24-25,2010 at N T U
by Professor Takashi Ito
Key Inequality
If we take in the above inequality we have
z i
2
1 1 2inf
2 p PL
p xR i x i R i
March 24-25,2010 at N T U
We can easily prove the above results when we use this inequality
2
1inf 0p P
L
R i p xx i
Theorem Let : 0
nP closelinear hull of x i n
21 Lx i P
2 2LP L
We can apply this theorem to characterize N-extreme measures.
March 24-25,2010 at N T U
Proof of Theorem
trivial
Proof of We shall prove 22 Lx i P
2n Lx i P which implies
2 2
2
1 1p xd p x d
x i x ix i
March 24-25,2010 at N T U
p x x i r x c
p x c
r xx i x i
2
p xq x d
x i
2
2 4p x
q x dx i
By Minkowskii’s inequality
March 24-25,2010 at N T U
Proof of Theorem
closed linear hull of : 1, 2,n
x i n 2L
In order to prove that
we can only notice Hahn-Banach theorem that
0, 1,2,n
f xd n
x i
imply 0, . ( )f a e
In fact, for any complex
10
0n
nn
f x f xd z x
x z x i
z
March 24-25,2010 at N T U
Proof of Theorem
N-extremal measure
Achiezer defined N-extreme measure
V
1) Indeterminate
2) Polynomial dense in
: k kV x d x d V Is one point set
determinate
indeterminatecontains more than two points
2L is N-extremal
March 24-25,2010 at N T U
Characterization by Geometry Meaning
Is N-extremal if and only if
iP Is co-dimension one in 2L
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iP x i p p P
Characterization of N-extremal measure N-extremeness implies the measure
is atomic ( due to L. de Brange )
B
B
n
B the set of zeros of the entire function B z
i.e. discrete or isolated point set
March 24-25,2010 at N T U
Entire Function Theorem . (Borichev,Sodin) A positive measure is N-extremal if and only if for some B(z) and its zero set , we have
(1)
(2) ( )
(3) ( )
B
B
n
2 2
1
1B B
2
1
F F
F B
March 24-25,2010 at N T U
2 2LP L
2 2LP L
B
1
0A B
we can find an entire function A z
of exponential type 0 such that
March 24-25,2010 at N T U
A.Borichev, M.Sodin,
The Hamburger Moment Problem and Weighted Polynomial Approximation on the Discrete Subsets of the Real Line, J.Anal.Math.76(1998),219-264
Conclusion We saw a connection between moment problem theory and prediction theory. Much remains to be done to clarify the statistical content of the whole subject.
March 24-25,2010 at N T U
Thank you
March 24-25,2010 at N T U