The tail behaviour of a random sum of subexponential random variables and vectors

Abstract

Let \(\left\{ X,X_{i},i=1,2,...\right\} \) denote independent positive random variables having common distribution function (d.f.) F(x) and, independent of X, let ν denote an integer valued random variable. Using X 0=0, the random sum Z=∑ i=0 ν X i has d.f. \(G(x)=\sum_{n=0}^{\infty }\Pr\{\nu =n\}F^{n\ast }(x)\) where F n∗(x) denotes the n-fold convolution of F with itself. If F is subexponential, Kesten’s bound states that for each e>0 we can find a constant K such that the inequality $$ 1-F^{n\ast }(x)\leq K(1+\varepsilon )^{n}(1-F(x))\, , \qquad n\geq 1,x\geq 0 \, , $$ holds. When F is subexponential and E(1 +e) ν <∞, it is a standard result in risk theory that G(x) satisfies $$ 1 - G{\left( x \right)} \sim E{\left( \nu \right)}{\left( {1 - F{\left( x \right)}} \right)},\,\,x \to \infty \,\,{\left( * \right)} $$ In this paper, we show that (*) holds under weaker assumptions on ν and under stronger conditions on F. Stam (Adv. Appl. Prob. 5:308–327, 1973) considered the case where \( \overline{F}(x)=1-F(x)\) is regularly varying with index –α. He proved that if α>1 and \(E{\left( {\nu ^{{\alpha + \varepsilon }} } \right)} -1\), then again Eν<∞ is sufficient. The proofs of the results rely on deriving bounds for the ratio \(\overline{F^{n\ast }}(x)/\overline{F} (x)\). In the paper, we also consider (*) in the special case where X is a positive stable random variable or has a compound Poisson distribution derived from such a random variable and, in this case, we show that for n≥2, the ratio \(\overline{F^{n\ast }}(x)/\overline{F}(x)\uparrow n\) as x↑∞. In Section 3 of the paper, we briefly discuss an extension of Kesten’s inequality. In the final section of the paper, we discuss a multivariate analogue of (*).