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June, 1981 A Method of Approximating Expectations of Functions of Sums of Independent Random Variables
Michael J. Klass
Ann. Probab. 9(3): 413-428 (June, 1981). DOI: 10.1214/aop/1176994415

Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent random variables with $S_n = \sum^n_{i = 1} X_i$. Fix $\alpha > 0$. Let $\Phi(\cdot)$ be a continuous, strictly increasing function on $\lbrack 0, \infty)$ such that $\Phi(0) = 0$ and $\Phi(cx) \leq c^\alpha\Phi(x)$ for all $x > 0$ and all $c \geq 2$. Suppose $a$ is a real number and $J$ is a finite nonempty subset of the positive integers. In this paper we are interested in approximating $E \max_{j \in J} \Phi(|a + S_j|)$. We construct a number $b_J(a)$ from the one-dimensional distributions of the $X$'s such that the ratio $E \max_{j \in J} \Phi(|a + S_j|)/\Phi(b_J(a))$ is bounded above and below by positive constants which depend only on $\alpha$. Bounds for these constants are given.

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Michael J. Klass. "A Method of Approximating Expectations of Functions of Sums of Independent Random Variables." Ann. Probab. 9 (3) 413 - 428, June, 1981. https://doi.org/10.1214/aop/1176994415

Information

Published: June, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0463.60023
MathSciNet: MR614627
Digital Object Identifier: 10.1214/aop/1176994415

Subjects:
Primary: 60G50
Secondary: 60E15 , 60J15

Keywords: $K$-function , approximation of expectations , approximation of integrals , expectations , Sums of independent random variables , tail $\Phi$-moment , truncated expectation , truncated mean , truncated second moment

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 3 • June, 1981
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