Approximations for compound Poisson and Pólya processes

Abstract

Suppose Xi ≧0 are i.i.d., i = 1, 2, ···. We derive a saddlepoint approximation for P{∑N(t) k=1 X k> y} as y→∞ and t is fixed, where N(t), t≧0, is either a Poisson or a Pólya process. These results are then compared and contrasted with the well-known Esscher approximation.