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.