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Hardy's condition in the moment problem for probability distributions

Lookup NU author(s): Dr Jordan Stoyanov

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Abstract

In probabilistic terms Hardy condition is written as follows: $E[e^{c{\sqrt X}}]<\infty$, where $X$ is a nonnegative random variable and $c>0$ a constant. If this holds then all moments of $X$ are finite and the distribution of $X$ is uniquely determined by the moments. This condition, based on two papers by G.H. Hardy (1917/1918), is weaker than Cram\'{e}r condition requiring the existence of a moment generating function of $X$. We elaborate Hardy condition and show that the constant $\frac12$ (square root) is the best possible for the moment determinacy of $X$. A characterization of Hardy condition in terms of the moments of $X$ is established. A corollary is derived in the multi-dimensional case.}

Publication metadata

Author(s): Stoyanov J, Lin GD

Publication type: Article

Publication status: Published

Journal: Theory of Probability and Its Applications

Year: 2012

Volume: 57

Issue: 4

Pages: 811-820

Print publication date: 01/12/2012

ISSN (print): 0040-585X

ISSN (electronic): 1095-7219

Publisher: Society for Industrial and Applied Mathematics

URL: http://dx.doi.org/10.4213/tvp4485

DOI: 10.4213/tvp4485

Notes: SIAM edition comes soon after the original Russian edition of the journal. The paper is in English.

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