ON THE MOMENT DETERMINACY OF PRODUCTS OF
NON-IDENTICALLY DISTRIBUTED RANDOM VARIABLES
Gwo Dong Lin
Jordan Stoyanov
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.
2000 AMS Mathematics Subject Classification: Primary: 60E05, 44A60; Secondary:
62E10.
Keywords and phrases: Product of random variables, Stieltjes moment problem,
Hamburger moment problem, Cramé r’s condition, Carleman’s condition, Krein’s condition,
Hardy’s condition.