On the extreme points of moments sets

Necessary and sufficient conditions for a measure to be an extreme point of the set of measures on a given measurable space with prescribed generalized moments are given, as well as an application to extremal problems over such moment sets; these conditions are expressed in terms of atomic partitions of the measurable space. It is also shown that every such extreme measure can be adequately represented by a linear combination of k Dirac probability measures with nonnegative coefficients, where k is the number of restrictions on moments; moreover, when the measurable space has appropriate topological properties, the phrase “can be adequately represented by” here can be replaced simply by “is”. Applications to specific extremal problems are also given, including an exact lower bound on the exponential moments of truncated random variables, exact lower bounds on generalized moments of the interarrival distribution in queuing systems, and probability measures on product spaces with prescribed generalized marginal moments.

[1]  I. Pinelis.  Exact Rosenthal-type bounds , 2013, 1304.4609.

[2]  Wassily Hoeffding,et al.  The Extrema of the Expected Value of a Function of Independent Random Variables , 1955 .

[3]  Sergey Utev,et al.  Exact Exponential Bounds for Sums of Independent Random Variables , 1990 .

[4]  I. Pinelis.  Exact lower bounds on the exponential moments of Winsorized and truncated random variables , 2010, 1001.2901.

[5]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[6]  Ward Whitt,et al.  Untold Horrors of the Waiting Room: What the Equilibrium Distribution Will Never Tell About the Queue-Length Process , 1983 .

[7]  Chee-Keng Yap,et al.  Fundamental problems of algorithmic algebra , 1999 .

[8]  H. Richter.  Parameterfreie Abschätzung und Realisierung von Erwartungswerten , 1957 .

[9]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[10]  Iosif Pinelis,et al.  An asymptotically Gaussian bound on the Rademacher tails , 2010, 1007.2137.

[11]  Gerhard Winkler,et al.  Integral representation in the set of solutions of a generalized moment problem , 1979 .

[12]  Alan F. Karr,et al.  Extreme Points of Certain Sets of Probability Measures, with Applications , 1983, Math. Oper. Res..

[13]  G. Bennett.  Probability Inequalities for the Sum of Independent Random Variables , 1962 .

[14]  I. Pinelis.  Tchebycheff systems and extremal problems for generalized moments: a brief survey , 2011, 1107.3493.

[15]  Flemming Topsøe,et al.  Topology and Measure , 1970 .

[16]  C. A. Rogers,et al.  Representation Theorems for Distribution Functions , 1958 .

[17]  A. E. Eckberg,et al.  Sharp Bounds on Laplace-Stieltjes Transforms, with Applications to Various Queueing Problems , 1977, Math. Oper. Res..

[18]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[19]  J. Kemperman.  On the Role of Duality in the Theory of Moments , 1983 .

[20]  Wolfgang Adamski.  Complete spaces and zero-one measures , 1976 .

[21]  Gerhard Winkler,et al.  Extreme Points of Moment Sets , 1988, Math. Oper. Res..

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

[23]  I. Pinelis.  Exact Lower Bounds on the Exponential Moments of Truncated Random Variables , 2011, Journal of Applied Probability.

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

[25]  I. Pinelis.  On the Bennett–Hoeffding inequality , 2009, 0902.4058.

[26]  R. Phelps.  Lectures on Choquet's Theorem , 1966 .