Optimal Upper and Lower Bounds for the Upper Tails of Compound Poisson Processes

A compound Poisson process is of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$S_{N_\lambda } = \sum\nolimits_{j = 1}^{N_\lambda } {Z_j }$$ \end{document} where Z, Z1, Z2, are arbitrary i.i.d. random variables and Nλ is an independent Poisson random variable with parameter λ. This paper identifies the degree of precision that can be achieved when using exponential bounds together with a single truncation to approximate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$P(S_{N_\lambda } \geqslant \lambda a)$$ \end{document}. The truncation level introduced depends only on λ and Z and not on the overall exceedance level λa.