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)

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Continuous Time Case

Conditions of deterministic 　

2

0

logw d

2

log

1

wd

sd w d d

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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

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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

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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

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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

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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)

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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

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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

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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

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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

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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

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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

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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

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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).

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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

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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

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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

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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

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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)

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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.

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（ 4 ）　　　　　　　 is dense in P 2L 21d x x d x

(5)

is dense in

iP x i p p P

2L

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Is determinate(3)

(1) (2) (3) (4) (5) are equivalent

Equivalence

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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

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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.

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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

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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

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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

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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

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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

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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

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2 2LP L

2 2LP L

B

1

0A B

we can find an entire function A z

of exponential type 0 such that

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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.

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Thank you 　

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