The Product of Independent Random Variables with Regularly Varying Tails

Abstract

A distribution μ is said to have regularly varying tail with index −α (α≥0) if lim  _x→∞μ(kx,∞)/μ(x,∞)=k ^−α for each k>0. Let X and Y be independent positive random variables with distributions μ and ν, respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of μ and ν. It is known that D(α) (the class of distributions on (0,∞) that have regularly varying tails with index −α) is closed under MS convolution. This paper deals with decomposition problem of distributions in D(α) related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)=∑ _k=0 ⁿ⁻¹ b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n−1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions μ and ν such that neither μ nor ν belongs to D(α) (α>0) and their MS convolution belongs to D(α).