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

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. $G(x)=sum_{n=0}^{infty }Pr{nu =n}F^{nast }(x)$ where F ^n?(x) denotes the n-fold convolution of F with itself. If F is subexponential, Kesten’s bound states that for each ε>0 we can find a constant K such that the inequality $$ 1-F^{nast }(x)leq K(1+varepsilon )^{n}(1-F(x)), , qquad ngeq 1,xgeq 0 , , $$ holds. When F is subexponential and E(1 +ε) ^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)} < infty $ , then relation (*) holds. For 0<α<1, it is sufficient that Eν<∞. In this paper we consider the case where $overline{F}(x)$ is an O-regularly varying subexponential function. If the lower Matuszewska index $beta (overline{F})<-1$ , then the condition ${text{E}}{left( {nu ^{{{left| {beta {left( {overline{F} } right)}} right|} + 1 + varepsilon }} } right)} < infty$ is sufficient for (*). If $beta (overline{F} )>-1$ , then again Eν<∞ is sufficient. The proofs of the results rely on deriving bounds for the ratio $overline{F^{nast }}(x)/overline{F} (x)$ . 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 $overline{F^{nast }}(x)/overline{F}(x)uparrow n$ 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 (*).