## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 3 (1981), 413-428.

### A Method of Approximating Expectations of Functions of Sums of Independent Random Variables

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 3 (1981), 413-428.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994415

**Digital Object Identifier**

doi:10.1214/aop/1176994415

**Mathematical Reviews number (MathSciNet)**

MR614627

**Zentralblatt MATH identifier**

0463.60023

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 60E15: Inequalities; stochastic orderings 60J15

**Keywords**

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

#### Citation

Klass, Michael J. A Method of Approximating Expectations of Functions of Sums of Independent Random Variables. Ann. Probab. 9 (1981), no. 3, 413--428. doi:10.1214/aop/1176994415. https://projecteuclid.org/euclid.aop/1176994415