Infinite divisibility of products and quotients of i.i.d. random variables

Given independent and identically distributed (i.i.d) random variables X and Y, we consider the infinite divisibility of XY and X/Y when X is (is not) infinitely divisible.For example, we prove that the product and quotient of two i.i.d. standard Cauchy randomvariables are infinitely divisible, and that the product of two i.i.d. Poisson random variables as well as the quotient of two i.i.d. Pareto random variables are not infinitely divisible. We also consider the possible infinite divisibility of 1/X

A theorem of Deny with applications to characterization problems

A theorem of Deny is stated and applications to certain characterization problems are indicated. A martingale proof of Deny’s theorem is given for a countable Abelian group