Open Access
Translator Disclaimer
July 1997 Approximation of partial sums of arbitrary i.i.d. random variables and the precision of the usual exponential upper bound
Marjorie G. Hahn, Michael J. Klass
Ann. Probab. 25(3): 1451-1470 (July 1997). DOI: 10.1214/aop/1024404520

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.

Citation

Download Citation

Marjorie G. Hahn. Michael J. Klass. "Approximation of partial sums of arbitrary i.i.d. random variables and the precision of the usual exponential upper bound." Ann. Probab. 25 (3) 1451 - 1470, July 1997. https://doi.org/10.1214/aop/1024404520

Information

Published: July 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0891.62007
MathSciNet: MR1457626
Digital Object Identifier: 10.1214/aop/1024404520

Subjects:
Primary: 60E15, 60F10, 62E17
Secondary: 60F05, 62E20

Rights: Copyright © 1997 Institute of Mathematical Statistics

JOURNAL ARTICLE
20 PAGES


SHARE
Vol.25 • No. 3 • July 1997
Back to Top