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).