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
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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
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Stationary Stochastic Process PredictionTheory Truncated Moment Problem Infinite Moment Problem Polynomial Dense N-extreme Measure Conclusion
Topics ,Key words
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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
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Discrete Time Case( Time Series)
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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
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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
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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
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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
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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
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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
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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
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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
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We can have an exact prediction from the past
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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(1) (2) (3) (4) (5) are equivalent
Equivalence
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has been proved by Riesz, Pollard and Achiezer
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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
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by Professor Takashi Ito
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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
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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
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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
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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
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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
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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
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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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