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
Topics ,Key words







Stationary Stochastic Process
PredictionTheory
Truncated Moment Problem
Infinite Moment Problem
Polynomial Dense
N-extreme Measure
Conclusion
Stationary Stochastic Sequences
Discrete Time Case（Time Series)

 , F , P 
Let
Probability space
X n , n  0, 1, 2,
Random variables with time variable n
EX n   X n dP  0,
 X n , X m P  EX n X m    n  m

weakly stationary
 k  
2
ik 
e
 d   
Spectral representation
0

Positive Borel Measure
March 24-25,2010 at N T U
Stationary stochastic process
Continuous Time Case
X  t ,   : t   ,   ,  
EX  t ,     X  t ,  P  d   0

  t  s   EX  t ,   X  s,     X  t  , X  s  P
 t  

i t
e
 d   

Spectral representation
(Bochner’s theorem)
March 24-25,2010 at N T U
Conditions of deterministic
2
 log w    d   
X n
is deterministic
0



log w   
1 
2
d   
 X  t 
is deterministic
d   w    d   d s
Conformal mapping from the unit circle to upper half plane
March 24-25,2010 at N T U
Transform the probability space into the
Discrete time case
function space
2
 , F , P 
L T ,  
Space of random variables
Space of square
with finite variance
summable functions
a0 X 0  a1 X 1  ...  an X n  a0  a1 z  ...  an z n
X k   
2
ik 
ik 
k
e
Z

,


e

z
, k  0,1, 2,



,n
0
2
 X n , X m  P    n  m    e i  n  m   d       z n , z m  L T ,  
2
0
March 24-25,2010 at N T U
isometry
Discrete time case
a0 X 0  a1 X 1  ...  an X n  a0  a1 z  ...  an z
n
Y    f  z 
2
E Y  a0 X 0  a1 X 1...  an X n 
2
 f z  a
0
 a1 z...  an z
n 2
d
0
Y  a0 X 0  a1 X 1  ...  an X n
P
 f  z   a0  a1 z  ...  an z n
L2 T ,  
Statistical Estimation error = Approximation error
March 24-25,2010 at N T U
Kolmogorov-Szego’s Theorem of
Discrete time
Prediction
Szegö’s Theorem:(Kolmogorov refound)
2
inf
a1 , a2 ,
2
 1   a z   w    d   exp  logw    d 
1
0
Kolmogorov’s Theorem
2
inf
a1 , a2 ,
 1 a z  
1
0
2
d   inf
a1 , a2 ,
2
2
 1 a z   w  d
1
0
d   w    d   d s , d  : Lebesgue measure
March 24-25,2010 at N T U
Prediction Error
inf E X m  a1 X m1  a2 X m2 
ak
2

 exp   log w    d   , if
0

2
2

 exp   log w    d  
0

2
 log w    d   
0
indeterministic
2
 0,
if
 log w    d   
0
deterministic
March 24-25,2010 at N T U
History

A.N.Kolmogorov , Interpolation and Extrapolation of Stationary
Sequences, Izvestiya AN SSSR (seriya matematicheskaya),5 (1941), 314
(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
Wiener
Hilbert Space (astract Math.)
Fourier Analysis
(Engineering sense)
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Szegö’s Alternative
Continuous time

Either



log w   
1  2
indeterministic
d   
and
L2     Z  0 
Z T
T 0
where
w  d
Absolute continuous part of
d   
Z ab  linearspanof eit , a  t  b, in L2   
March 24-25,2010 at N T U
or else
Deterministic case
Continuous time



log w   
1 
2
d   
then
L2     Z  0 
Z T
T 0
Z ab  linearspanof eit , a  t  b, in L2   
We can have an exact prediction from the past
March 24-25,2010 at N T U
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,1976
Dover edition,2008
March 24-25,2010 at N T U
Problem of Krein
Finite Prediction
From finite segment of past
X  t ,   , 2T  t  0,
Predict the future value
X t,   , t  0
Compute the projection of
i  t T 
on
e
Z T  span of eit t  T
Krein’s idea=Analyze String and spectral function
Moment Problem Technique ( see Dym- Mckean book in detail)
March 24-25,2010 at N T U
Moment Problem
 k  
2
e
ik 
d   z d  ,
0
k
T   z  e i 
T
  0 ,  1 ,   2 ,

uniquely determined
  0 ,  1 ,   2 ,
 N

indeterminated
March 24-25,2010 at N T U
Representing measure

is called the representing measure of
if
 k  
  0 ,  1 ,   2  ,
 N
2
ik 
e
 d   
0
M    0  ,  1 ,
2

,   N        k    eik  d     , k  0,1, 2,

0

N

a set of representation measures( convex set)
We particularly have an interest to find
the extreme points of
M    0 ,  1 ,
,   N 
March 24-25,2010 at N T U
Truncated Moment Problem
  0 ,  1 ,   2 ,
N
 N
N
   j  k  a j ak  0,
Positive definite
j 0 k 0
2
N
for any
a0 , a1 ,
, aN
such taht
a
j 0
j
0
Find representing measures of
which moments are   0 ,  1 ,   2 ,   N 
And characterize the totality of
representation measures
March 24-25,2010 at N T U
Properties of Extreme Points

M    0 ,  1 ,
is an exｔreme point of conves set
L1  d    linear span{z k k  0, 1, 2,
Polynomial dense in

,   N 
 N , z  ei }
L1  d    L2  d  
is the representing measure for a singular extension of
  0 ,  1 ,   2 ,
 N
March 24-25,2010 at N T U
Singularly positive definite sequence
Trucated Moment Problem

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
Singular positive definite sequence
c0 , c1,
, cM , cM 1
c0 , c1,
c0 , c1,
, cM 
Is singular positive definite
is positive definite
, cM , cM 1 is nonegative definite but positive definite
March 24-25,2010 at N T U
Theorem: extreme measures

is an extreme point of
2
dk 
e
ik 
M c0 , c1 ,
, cN 
d  , k  0,1, 2
M 1
0
d0 , d1,
c0 , c1,
, dM , dM 1
, cN 
is singular extenstion of
N  M  2N
(i.e. dk  ck ,0  k  N )
March 24-25,2010 at N T U
Extreme points of representing measures
z
n
, zm  
2

e
i  n  m 
d   
0

Let
N
EM  z    Pk  Pk  z 
P0  z  , P1  z  ,
k 0
Orthonormal polynomials
Singularly Positive Sequence
c0 , c1 ,
determines uniquely measure as
, cN 
N 1
 
k 1
where
ak , k  1, 2,
ak , k  1, 2,
N 1
PN  z 
N 1
1
E
ak
N
2
a
k
PN 1  z 
are zeros of a polynomial
simple roots on the unit circle ak  1
.
March 24-25,2010 at N T U
Hamburger Moment Problem
Infinite Moment Problem


(*) sk 
x k d   x , k  0,1, 2...,

where
sk , k  0,1, 2,
sk 

has infinite support
Find

is a moment sequence of
satisfying (*)

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Achiezer : Classical Moment Problem
March 24-25,2010 at N T U
Riesz’s criterion

R  z   sup p  z  p
pP
(1)
For some
z0 
(1’） For any
L2   

1
\ , R  z0   
z
\ ,
R  z   
March 24-25,2010 at N T U
The Logarithmic Integral

(2)



log  R  x 
1 x
2
dx  
This is a common formula
which appears in the
moment problem and the
prediction theory.
March 24-25,2010 at N T U

(3)
（4）
P
Is determinate
is dense in
L 
2
d   x   1  x 2  d   x 
(5)
Pi   x  i  p p  P
is dense in
L2   
March 24-25,2010 at N T U
Equivalence
(1) (2) (3) (4) (5) are equivalent
has been proved by Riesz, Pollard and Achiezer
March 24-25,2010 at N T U
Important Inequality
by Professor Takashi Ito
1
1
1
 inf
 p  x
1  z R  z  pP x  z

L2   
1 z
1
Im z R  z 
d   x   1  x 2  d   x 
P
polynomials
March 24-25,2010 at N T U
Key Inequality

z i
If we take
in the above inequality
we have
1
1
2
 inf
 p  x

p

P
2 R  i 
x i
R  i 
L2   
We can easily prove the above results when we use
this inequality
1
R  i     inf
 p  x
pP x  i
0
L2   
March 24-25,2010 at N T U
Theorem

Let
P  closelinear hull of
 x  i
P
1
L2   
P
 x  i  : n  0
n
L2   
 L2   
We can apply this theorem to characterize N-extreme measures.
March 24-25,2010 at N T U
Proof of Theorem

trivial
We shall prove  x  i   P
2
Proof of
which implies
2
1
  x  i
2
 x  i
n
L2   
P
L2   
p  x
1

d  
 p  x d   
 x  i
 x  i
2
March 24-25,2010 at N T U
Proof of Theorem
p  x   x  i  r  x  c
p  x
c
 r  x 
x i
 x  i
p  x
 q  x d   
x i
2

p  x
  x  i
2
2
 q  x  d   4
By Minkowskii’s inequality
March 24-25,2010 at N T U
Proof of Theorem
In order to prove that
closed linear hull of
 x  i 
n
: n  1, 2,

 L2   
we can only notice Hahn-Banach theorem that
f  x
  x  i
n
d   0, n  1, 2,
imply
In fact, for any complex

f  x
f  0, a.e (  )
z

f  x 
n
d   
 z  x  0
n 1 


xz
x

i
n 0





March 24-25,2010 at N T U
N-extremal measure

V   :  x k d   x k d 
V
V


Is one point set
determinate
contains more than two points
indeterminate
Achiezer defined N-extreme measure
 is N-extremal
1) Indeterminate
2) Polynomial dense in L   
2
March 24-25,2010 at N T U
Characterization by Geometry Meaning

Pi
Is N-extremal if and only if
Is co-dimension one in
L2   
Pi   x  i  p p  P
March 24-25,2010 at N T U
Characterization of N-extremal measure
N-extremeness implies the measure
is atomic ( due to L. de Brange )


  


 B
B


 B

   
n
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
B

(1)



 


 B
 B

(2)


 B

(3)
1
  B    1   


 F
2
1
  F    
2
2

 (
(
P
P
   
n
L2   
L2   
 L2    )
 L2   
)
F  B
March 24-25,2010 at N T U
we can find an entire function
A z 
of exponential type 0 such that
1
 
0
A    B   
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
March 24-25,2010 at N T U
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