The Product of Independent Random Variables with Regularly Varying Tails

Takaaki Shimura ()
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Takaaki Shimura: The Institute of Statistical Mathematics

A chapter in Recent Developments in Infinite-Dimensional Analysis and Quantum Probability, 2001, pp 411-432 from Springer

Abstract: 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).

Keywords: primary 60E05; secondary 60E07; 60F05; tail of distribution; regularly varying function; slowly varying function; product of independent random variables (search for similar items in EconPapers)
Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-010-0842-6_29

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DOI: 10.1007/978-94-010-0842-6_29

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