A Method of Approximating Expectations of Functions of Sums of Independent Random Variables
Other literature type
English
OPEN
Klass, Michael J.
(1981)
 Publisher: The Institute of Mathematical Statistics

Journal:
(issn: 00911798)

Related identifiers:
doi: 10.1214/aop/1176994415

Subject:
Sums of independent random variables  expectations  truncated mean  truncated expectation  truncated second moment  tail $\Phi$moment  $K$function  approximation of expectations  approximation of integrals  60G50  60E15  60J15
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 onedimensional 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.