Approximations for compound Poisson and Pólya processes

Suppose X i ≧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.