On the Moment Determinacy of Products of Non-identically Distributed Random Variables

TL;DR

This paper investigates the conditions under which the products of independent, non-identically distributed random variables are uniquely determined by their moments, covering various cases including generalized gamma distributions.

Contribution

It introduces new checkable conditions and results for the moment (in)determinacy of products of diverse independent random variables, extending existing theories.

Findings

Product of exponential and inverse Gaussian is moment determinate.

Product of exponential and normal is moment indeterminate.

Provides characterization for generalized gamma distribution products.

Abstract

We show first that there are intrinsic relationships among different conditions, old and recent, which lead to some general statements in both the Stieltjes and the Hamburger moment problems. Then we describe checkable conditions and prove new results about the moment (in)determinacy for products of independent and non-identically distributed random variables. We treat all three cases when the random variables are nonnegative (Stieltjes case), when they take values in the whole real line (Hamburger case), and the mixed case. As an illustration we characterize the moment determinacy of products of random variables whose distributions are generalized gamma or double generalized gamma all with distinct shape parameters. Among other corollaries, the product of two independent random variables, one exponential and one inverse Gaussian, is moment determinate, while the product is moment indeterminate for the cases: one exponential and one normal, one chi-square and one normal, and one inverse Gaussian and one normal.