The truncated moment problem on N0

Abstract We find necessary and sufficient conditions for the existence of a probability measure on N 0 , the nonnegative integers, whose first n moments are a given n -tuple of nonnegative real numbers. The results, based on finding an optimal polynomial of degree n which is nonnegative on N 0 (and which depends on the moments), and requiring that its expectation be nonnegative, generalize previous results known for n = 1 , n = 2 (the Percus–Yamada condition), and partially for n = 3 . The conditions for realizability are given explicitly for n ≤ 5 and in a finitely computable form for n ≥ 6 . We also find, for all n , explicit bounds, in terms of the moments, whose satisfaction is enough to guarantee realizability. Analogous results are given for the truncated moment problem on an infinite discrete semi-bounded subset of R .

[1]  Masami Yamada,et al.  Geometrical Study of the Pair Distribution Function in the Many-Body Problem , 1961 .

[2]  J. Lebowitz,et al.  Necessary and sufficient conditions for realizability of point processes , 2009, 0910.1710.

[3]  M. Laurent.  Sums of Squares, Moment Matrices and Optimization Over Polynomials , 2009 .

[4]  E. Caglioti,et al.  Translation invariant realizability problem on the $d-$dimensional lattice: an explicit construction , 2015, 1510.02954.

[5]  W. J. Studden,et al.  Tchebycheff Systems: With Applications in Analysis and Statistics. , 1967 .

[6]  L. Fialkow,et al.  THE TRUNCATED K-MOMENT PROBLEM: A SURVEY , 2015 .

[7]  Raúl E. Curto,et al.  An analogue of the Riesz–Haviland theorem for the truncated moment problem , 2008 .

[8]  J. Lasserre.  Moments, Positive Polynomials And Their Applications , 2009 .

[9]  Endre Boros,et al.  Closed Form Two-Sided Bounds for Probabilities that At Least r and Exactly r Out of n Events Occur , 1989, Math. Oper. Res..

[10]  Josef Teichmann,et al.  The proof of Tchakaloff’s Theorem , 2005 .

[11]  Seymour M. Kwerel.  Most Stringent Bounds on Aggregated Probabilities of Partially Specified Dependent Probability Systems , 1975 .

[12]  J. Lebowitz,et al.  Point processes with speci?ed low order correlations , 2006 .

[13]  L. Shapley,et al.  Geometry of Moment Spaces , 1953 .

[14]  E. R. Speer,et al.  Realizability of Point Processes , 2007 .

[15]  Seymour M. Kwerel.  Bounds on the probability of the union and intersection of m events , 1975, Advances in Applied Probability.

[16]  J. Shohat,et al.  The problem of moments , 1943 .

[17]  A. Ivic.  Sums of squares , 2020, An Introduction to 𝑞-analysis.

[18]  András Prékopa,et al.  Lower and upper bounds for the probability that at least r and exactly r out of n events occur , 2002 .

[19]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[20]  M. Kreĭn,et al.  The Markov Moment Problem and Extremal Problems , 1977 .

[21]  András Prékopa,et al.  The discrete moment problem and linear programming , 1990, Discret. Appl. Math..