On the behavior of the product of independent random variables

Abstract

For two independent non-negative random variables X and Y , we treat X as the initial variable of major importance and Y as a modifier (such as the interest rate of a portfolio). Stability in the tail behaviors of the product compared with that of the original variable X is of practical interests. In this paper, we study the tail behaviors of the product XY when the distribution of X belongs to the classes L and S, respectively. Under appropriate conditions, we show that the distribution of the product XY is in the same class as X when X belongs to class L or S, in other words, classes L and S are stable under some mild conditions on the distribution of Y . We also show that if the distribution of X is in class L(γ) (γ > 0) and continuous, then the product XY is in L if and only if Y is unbounded.

From light tails to heavy tails through multiplier

“…Since they were introduced by Chistyakov (1964) and Chover et al (1973a,b) these classes have been extensively investigated and have been applied to many fields of probability theory. It is well known that the constant c in (1.2) is equal to Pakes (2004), Shimura and Watanabe (2005), Tang (2006a), Su and Chen (2006), and Foss and Korshunov (2007), among others. If γ = 0 then relations (1.1) and (1.2) describe the well-known long-tailed distribution class L(0) and subexponential distribution class S(0), respectively.…”

Modelling heavy-tails with phase-type scale mixture distributions

“…Hence, it is plausible that if Π G is close enough to the integrated tail distribution F of the claim sizes, then we can use ψ Π G (u) as an approximation for ψ F (u), the ruin probability of a Cramér-Lundberg process having claim size distribution F . One of the key features of the class of phase-type scale mixtures is that if Π has unbounded support, then Π G is a heavy-tailed distribution (Rojas-Nandayapa and Xie, 2017; Su and Chen, 2006;Tang, 2008), confirming the hypothesis that the class of phase-type scale mixtures is more appropriate for approximating tail-dependent quantities involving heavytailed distributions. In contrast, the class of classical phase-type distributions is light-tailed and approximations derived from this approach may be inaccurate in the tails (see also Vatamidou et al, 2014, for an extended discussion).…”

“…For instance, Su and Chen (2006) show that if two random variables S 1 and S 2 are such that the distribution of S 1 is in L(λ) with λ > 0 and S 2 has unbounded support, then the distribution of S 1 · S 2 is in L(0) (long-tailed), and thus heavy-tailed (see also Tang, 2008). If one further assumes that S 2 is Weibullian with parameter 0 < p 1, then Liu and Tang (2010) show that the product S 1 · S 2 is subexponential.…”

“…Notice that the distributions considered in Su and Chen (2006) correspond to distributions with log-tail probabilities decaying at a linear rate, i.e. − ln H 1 (s) = O(s), while the distributions in Arendarczyk and Dȩbicki (2011) have log-tail probabilities decaying at a power rate, i.e.…”

“…Asmussen and Albrecher, 2010). We collect conditions from Su and Chen (2006); Tang (2008) to determine the tail heaviness of phase-type scale mixtures and we provide a simple proof. That is, a phase-type scale mixture distribution is heavy-tailed if and only if the scaling distribution Π has unbounded support.…”