1. 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

2. Stationary Stochastic Process PredictionTheory Truncated Moment Problem Infinite Moment Problem Polynomial Dense N-extreme Measure Conclusion Topics,Key words

3. Stationary Stochastic Sequences Let Probability space Random variables with time variable n Spectral representation Positive Borel Measure weakly stationary March 24-25,2010 at N T U Discrete Time Case （ Time Series)

4. Stationary stochastic process Spectral representation (Bochner’s theorem) March 24-25,2010 at N T U Continuous Time Case

5. Conditions of deterministic March 24-25,2010 at N T U Conformal mapping from the unit circle to upper half plane is deterministic

6. Transform the probability space into the function space March 24-25,2010 at N T U Discrete time case Space of random variables with finite variance Space of square summable functions

7. isometry Statistical Estimation error = Approximation error March 24-25,2010 at N T U Discrete time case

8. Kolmogorov-Szego ’ s Theorem of Prediction Kolmogorov’s Theorem Szegö’s Theorem:(Kolmogorov refound) March 24-25,2010 at N T U Discrete time

9. Prediction Error March 24-25,2010 at N T U deterministic indeterministic

10. 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

11. Szeg ö ’s Alternative Either Absolute continuous part of and where indeterministic March 24-25,2010 at N T U Continuous time

12. or else Deterministic case then Continuous time March 24-25,2010 at N T U We can have an exact prediction from the past

13. 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

14. Problem of Krein Predict the future value on 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)

15. Moment Problem uniquely determined March 24-25,2010 at N T U indeterminated

16. Representing measure 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 a set of representation measures( convex set)

17. Truncated Moment Problem March 24-25,2010 at N T U for any such taht Positive definite Find representing measures of which moments are And characterize the totality of representation measures

18. Properties of Extreme Points is an ex ｔ reme point of conves set is the representing measure for a singular extension of March 24-25,2010 at N T U Polynomial dense in

19. 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

20. Singular positive definite sequence is positive definite is nonegative definite but positive definite Is singular positive definite March 24-25,2010 at N T U

21. Theorem: extreme measures is an extreme point of is singular extenstion of March 24-25,2010 at N T U

22. Extreme points of representing measures Let Singularly Positive Sequence determines uniquely measure as where are zeros of a polynomial March 24-25,2010 at N T U simple roots on the unit circle. Orthonormal polynomials

23. Hamburger Moment Problem Findsatisfying (*) is a moment sequence of March 24-25,2010 at N T U Infinite Moment Problem where has infinite support

24. Achiezer : Classical Moment Problem March 24-25,2010 at N T U

25. Riesz ’ s criterion (1 ’ ） (1) March 24-25,2010 at N T U For some For any

26. The Logarithmic Integral (2) This is a common formula which appears in the moment problem and the prediction theory. March 24-25,2010 at N T U

27. （ 4 ） is dense in (5) is dense in March 24-25,2010 at N T U Is determinate (3)

28. (1) (2) (3) (4) (5) are equivalent Equivalence March 24-25,2010 at N T U has been proved by Riesz, Pollard and Achiezer

29. Important Inequality polynomials March 24-25,2010 at N T U by Professor Takashi Ito

30. Key Inequality If we take in the above inequality we have March 24-25,2010 at N T U We can easily prove the above results when we use this inequality

31. Theorem Let We can apply this theorem to characterize N-extreme measures. March 24-25,2010 at N T U

32. Proof of Theorem trivial Proof of We shall prove which implies March 24-25,2010 at N T U

33. By Minkowskii’s inequality March 24-25,2010 at N T U Proof of Theorem

34. closed linear hull of In order to prove that we can only notice Hahn-Banach theorem that imply In fact, for any complex March 24-25,2010 at N T U Proof of Theorem

35. N-extremal measure Achiezer defined N-extreme measure 1)Indeterminate 2)Polynomial dense in Is one point set determinate indeterminatecontains more than two points is N-extremal March 24-25,2010 at N T U

36. Characterization by Geometry Meaning Is N-extremal if and only if Is co-dimension one in March 24-25,2010 at N T U

37. Characterization of N-extremal measure N-extremeness implies the measure is atomic ( due to L. de Brange ) the set of zeros of the entire function i.e. discrete or isolated point set March 24-25,2010 at N T U

38. 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) ( ) March 24-25,2010 at N T U

39. we can find an entire function 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

40. 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

41. Thank you March 24-25,2010 at N T U