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Hamburger moment problem for powers and products of random variables

Lookup NU author(s): Dr Jordan Stoyanov

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Abstract

We present new results on the Hamburger moment problem for probability distributions and apply them to characterize the moment determinacy of powers and products of i.i.d. random variables with values in the whole real line. Detailed proofs of all results are given followed by comments and examples. We also provide new and more transparent proofs of a few known results. E.g., we give a new and short proof that the product of three or more i.i.d. normal random variables is moment-indeterminate. The illustrations involve specific distributions such as the double generalized gamma (DGG), normal, Laplace and logistic. We show that sometimes, but not always, the power and the product of i.i.d. random variables (of the same odd 'order') share the same moment determinacy property. This is true for the DGG and the logistic distributions.The paper also treats two unconventional types of problems: products of independent random variables of different types and a random power of a given random variable. In particular, we show that the product of Laplace and logistic random variables, the product of logistic and exponential random variables, the product of normal and chi(2) random variables, and the random power Z(N), where Z similar to N and N is a Poisson random variable, are all moment-indeterminate. (C) 2013 Elsevier B.V. All rights reserved.


Publication metadata

Author(s): Stoyanov J, Lin GD, DasGupta A

Publication type: Article

Publication status: Published

Journal: Journal of Statistical Planning and Inference

Year: 2014

Volume: 154

Pages: 166-177

Print publication date: 01/11/2014

Online publication date: 16/11/2014

ISSN (print): 0378-3758

ISSN (electronic): 1873-1171

Publisher: Elsevier

URL: .http:/dx.doi.org/10.1016/j.jspi.2013.11.002

DOI: 10.1016/j.jspi.2013.11.002


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Funding

Funder referenceFunder name
Elsevier
Leverhulme Trust (UK)
NSC 102-2118-M-001-008-MY2National Science Council of ROC (Taiwan)

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