caa a22 4500 475839218 CHVBK 20180406123857.0 cr unu---uuuuu 170329e20000901xx s 000 0 eng 10.1023/A:1010714408233 doi (NATIONALLICENCE)springer-10.1023/A:1010714408233 Shimura Takaaki The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu Minato-ku, 106-8569, Tokyo, Japan. e-mail aut The Product of Independent Random Variables with Regularly Varying Tails [Elektronische Daten] [Takaaki Shimura] 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). Kluwer Academic Publishers, 2000 tail of distribution nationallicence regularly varying function nationallicence slowly varying function nationallicence product of independent random variables nationallicence Acta Applicandae Mathematica Kluwer Academic Publishers 63/1-3(2000-09-01), 411-432 0167-8019 63:1-3<411 2000 63 10440 https://doi.org/10.1023/A:1010714408233 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1010714408233 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Shimura Takaaki The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu Minato-ku, 106-8569, Tokyo, Japan. e-mail aut NATIONALLICENCE 773 0- Acta Applicandae Mathematica Kluwer Academic Publishers 63/1-3(2000-09-01), 411-432 0167-8019 63:1-3<411 2000 63 10440 Metadata rights reserved Springer special CC-BY-NC licence nationallicence SWISSBIB 387909605 BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer