Moment Problem and Density Questions Akio Arimoto

Moment Problem and Density QuestionsAkio Arimoto Mini-Workshop on Applied Analysis and Applied Probability March24-25,2010 at National Taiwan University March24-25,2010 at N T U

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

Let Stationary Stochastic Sequences Discrete Time Case（Time Series) Probability space Random variables with time variable n weakly stationary Spectral representation Positive Borel Measure March24-25,2010 at N T U

Stationary stochastic process Continuous Time Case Spectral representation (Bochner’s theorem) March24-25,2010 at N T U

Conditions of deterministic is deterministic is deterministic Conformal mapping from the unit circle to upper half plane March24-25,2010 at N T U

Transform the probability space into the function space Discrete time case Space of random variables with finite variance Space of square summable functions March24-25,2010 at N T U

isometry Discrete time case Statistical Estimation error = Approximation error March24-25,2010 at N T U

Kolmogorov-Szego’s Theoremof Prediction Discrete time Szegö’s Theorem:(Kolmogorov refound) Kolmogorov’s Theorem March24-25,2010 at N T U

Prediction Error indeterministic deterministic March24-25,2010 at N T U

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) March24-25,2010 at N T U

Szegö’sAlternative Continuous time • Either indeterministic and where Absolute continuous part of March24-25,2010 at N T U

or else Deterministic case Continuous time then We can have an exact prediction from the past March24-25,2010 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 March24-25,2010 at N T U

Problem of Krein Finite Prediction From finite segment of past Predict the future value Compute the projection of on Krein’s idea=Analyze String and spectral function Moment Problem Technique ( see Dym- Mckean book in detail) March24-25,2010 at N T U

Moment Problem uniquely determined indeterminated March24-25,2010 at N T U

Representing measure is called the representing measure of if a set of representation measures( convex set) We particularly have an interest to find the extreme points of March24-25,2010 at N T U

Truncated Moment Problem Positive definite such taht for any Find representing measures of which moments are And characterize the totality of representation measures March24-25,2010 at N T U

Properties of Extreme Points is an exｔreme point of conves set Polynomial dense in is the representing measure for a singular extension of March24-25,2010 at N T U

Singularlypositive definitesequence Trucated Moment Problem • Arimoto,Akio; Ito, Takashi, Singularly Positive Definite Sequences andParametrization of Extreme Points. Linear Algebra Appl. 239, 127-149(1996). March24-25,2010 at N T U

Singular positive definite sequence Is singular positive definite is positive definite is nonegative definite but positive definite March24-25,2010 at N T U

Theorem: extreme measures is an extreme point of is singularextenstion of March24-25,2010 at N T U

Extreme points of representing measures • Let Orthonormal polynomials Singularly Positive Sequence determines uniquely measure as where are zeros of a polynomial simple roots on the unit circle . March24-25,2010 at N T U

Hamburger Moment Problem Infinite Moment Problem where has infinite support Find satisfying (*) is a moment sequence of March24-25,2010 at N T U

Achiezer : Classical Moment Problem March24-25,2010 at N T U

Riesz’s criterion (1) For some (1’） For any March24-25,2010 at N T U

The Logarithmic Integral • (2) This is a common formula which appears in the moment problem and the prediction theory. March24-25,2010 at N T U

(3) Is determinate （4）　　　　　　　is dense in (5) is densein March24-25,2010 at N T U

(1) (2) (3) (4) (5) are equivalent Equivalence has been proved by Riesz, Pollard and Achiezer March24-25,2010 at N T U

Important Inequality by Professor Takashi Ito polynomials March24-25,2010 at N T U

Key Inequality • If we take in the above inequality we have We can easily prove the above results when we use this inequality March24-25,2010 at N T U

Theorem • Let We can apply this theorem to characterize N-extreme measures. March24-25,2010 at N T U

Proof of Theorem • trivial Proof of We shall prove which implies March24-25,2010 at N T U

Proof of Theorem By Minkowskii’s inequality March24-25,2010 at N T U

closed linear hull of Proof of Theorem In order to prove that we can only notice Hahn-Banach theorem that imply In fact, for any complex March24-25,2010 at N T U

N-extremal measure • Achiezerdefined N-extreme measure Is one point set determinate contains more than two points indeterminate • Indeterminate • Polynomial dense in is N-extremal March24-25,2010 at N T U

Characterization by Geometry Meaning Is N-extremal if and only if Is co-dimension one in March24-25,2010 at N T U

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 March24-25,2010 at N T U

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) ( ) Entire Function March24-25,2010 at N T U

we can find an entire function of exponential type 0 such that A.Borichev, M.Sodin, The Hamburger Moment Problem and Weighted Polynomial Approximation on the Discrete Subsets of the RealLine,J.Anal.Math.76(1998),219-264 March24-25,2010 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. March24-25,2010 at N T U

Thank you March24-25,2010 at N T U

Moment Problem and Density Questions Akio Arimoto