Optimal upper and lower bounds for the upper tails of compound Poisson processes (Q1266793)

A compund Poisson process is of the form \(S_{N_{\lambda}}= \sum_{j=1}^{N_{\lambda}}\) \(Z_j\) where \(Z, Z_1, Z_2, \dots\) are arbitrary i.i.d. random variables and \(N_{\lambda}\) is an independent Poisson random variable with parameter \(\lambda\). This paper identifies the degree of precision that can be achieved when using exponential bounds together with a single truncation to approximate \(P(S_{N_{\lambda}} \geq \lambda a)\). The truncation level introduced depends only on \(\lambda\) and \(Z\) and not on the overall exceedance level \(\lambda a\).