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

Let {X, X_i,i=1,2,...} 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)=∑_n=0^∞Pr{ν = n}F^n*(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^n*(x)≤ K(1+ε)ⁿ(1-F(x)), n≥ 1, x≥ 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(x ) ∼ E(ν)(1 - F(x), x → ∞ (*) 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 F̄(x)=1-F(x) is regularly varying with index -α. He proved that if α>1 and E(ν^α+ε) < ∞, then relation (*) holds. For 0<α<1, it is sufficient that Eν<∞. In this paper we consider the case where F̄(x) is an O-regularly varying subexponential function. If the lower Matuszewska index β (F̄)<-1, then the condition E(ν^{|β(F̄)|+1+ε)} < ∞ is sufficient for (*). If β(F̄ )>-1, then again Eν<∞ is sufficient. The proofs of the results rely on deriving bounds for the ratio F^n*̄(x)/F̄(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 F^n*̄(x)/ F̄(x)↑ 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 (*).