A note on convolutions of compound geometric distributions

Let G(x)G(x) be a compound geometric distribution function of a random variable SS, defined by G(x)=Pr(S≤x)=∑n=0∞(1−ϕ)ϕnF∗n(x) (0<ϕ<10<ϕ<1), and let A(x)A(x) be the DF of a random variable independent of SS. In this paper, we derive new results concerning stochastic comparisons of the function Kx(y)Kx(y) introduced by Willmot and Cai [Willmot, G.E., Cai, J. 2004. On application of residual lifetimes of compound geometric distributions. J. Appl. Probab. 41, 802–815], which is strongly related to the compound geometric convolution W(x)=G∗A(x)W(x)=G∗A(x). We also obtain asymptotic formulas for heavy-tailed distributions generalizing known results by Cai and Tang [Cai, J., Tang, Q., 2004. On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Probab. 41, 117–130]. Moreover, in the case of light-tailed distributions, we provide the moment generating function of Kx(y)Kx(y) at a point RR that satisfies a Lundberg type equation.