The Poisson Counting Argument: A Heuristic for Understanding What Makes a Poissonized Sum Large

Abstract

Let k be a fixed natural number and let Z, Z₁, Z₂,…, Z_k be arbitrary i.i.d. random variables. Hahn and Klass (1991a) shows that it is always possible to construct a function Fz,k (a) such that for some constant C_* > 0 (independent of Z, k, and a) and all real a,

$$C_{*}F_{Z,k}^{2}(a)\leq P(\sum_{j=1}^{k}Z_{j}\geq ka)\leq 2F_{z,k}(a).$$

Furthermore, Fz,k (a) is the usual exponential bound for Z modified by a single datadriven truncation. Thus, for arbitrary Z, k, and a, the correct order of magnitude of $LogP(\sum_{j=1}^{k}Z_{j}\geq ka)$ is obtainable essentially to within a factor of 2.