Approximation of partial sums of arbitrary i.i.d. random variables and the precision of the usual exponential upper bound

Abstract

This paper quantifies the degree to which exponential bounds can be used to approximate tail probabilities of partial sums of arbitrary i.i.d. random variables. The introduction of a single truncation allows the usual exponential upper bound to apply usefully whenever the summands are arbitrary i.i.d. random variables. More specifically, let n be a fixed natural number and let $Z, Z_1, Z_2, \dots, Z_n$ be arbitrary i.i.d. random variables. We construct a function $F_{Z, n} (a)$, derived from the probability of occurrence of one or more ‘‘large’’ summands plus an upper bound of exponential type, such that for some constant $C_* > 0$ (independent of $Z, n$ and $a$) and all real $a$,

$$C_* F_{Z,n}^2 (a) \leq P(\sum_{j=1}^n Z_j \geq na) \leq 2F_{Z,n} (a).$$

Furthermore, examples show that the upper and lower bounds are achievable.