Error Bounds for Exponential Approximations of Geometric Convolutions

Abstract

Define $Y_0$ to be a geometric convolution of $X$ if $Y_0$ is the sum of $N_0$ i.i.d. random variables distributed as $X$, where $N_0$ is geometrically distributed and independent of $X$. It is known that if $X$ is nonnegative with finite second moment, then as $p \rightarrow 0, Y_0/EY_0$ converges in distribution to an exponential distribution with mean 1. We derive an upper bound for $d(Y_0)$, the sup norm distance between $Y_0$ and an exponential with mean $EY_0$. This upper bound is $d(Y_0) \leq \operatorname{cp}$ for $0 < p \leq \frac{1}{2}$, where $c = EX^2/(EX)^2$. It is asymptotically $(p \rightarrow 0)$ tight. Also derived is a bound for $d(Y_0 + Z)$, where $Z$ is independent of $Y_0$.