Error Bounds for Exponential Approximations of Geometric Convolutions

Define Y₀ to be a geometric convolution of X if Y₀ is the sum of N₀ i.i.d. random variables distributed as X, where N₀ is geometrically distributed and independent of X. It is known that if X is nonnegative with finite second moment, then as p → 0, Y₀/EY₀ converges in distribution to an exponential distribution with mean 1. We derive an upper bound for d(Y₀), the sup norm distance between Y₀ and an exponential with mean EY₀. This upper bound is d(Y₀) ≤ cp for 0 < p ≤ 1/2, where c = EX²/(EX)². It is asymptotically (p → 0) tight. Also derived is a bound for d(Y₀ + Z), where Z is independent of Y₀.