Approximation of partial sums of arbitrary i. i. d. random variables and the precision of the usual exponential upper bound. (English) Zbl 0891.62007

Hahn, Marjorie G.; Klass, Michael J.

Ann. Probab. 25, No. 3, 1451-1470 (1997).

Summary: 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 \(a\), \[ C_*F^2_{Z,n} (a)\leq P \left(\sum^n_{j= 1} Z_j\geq na\right) \leq 2F_{Z,n}(a). \] Furthermore, examples show that the upper and lower bounds are achievable.

MSC:

62E17	Approximations to statistical distributions (nonasymptotic)
60E15	Inequalities; stochastic orderings
60F10	Large deviations
62E20	Asymptotic distribution theory in statistics
60F05	Central limit and other weak theorems

Keywords:

Esscher transform; approximation of exceedance levels; exponential upper bounds; local probability approximations; tail probabilities

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