AISM 54, 114-124

© 2002 ISM

## Expansions for the distributions of some normalized summations of random numbers of i.i.d. random variables

### Nan Wang^{1} and Wei Liu^{2}

^{1}Department of Social Statistics, University of
Southampton, Southampton, SO17 1BJ, U.K., e-mail: nw@socsci.soton.ac.uk

^{2}Department of Statistics, University of Central
Florida, Orlando, FL, U.S.A. and Department of Mathematics,University of Southampton,
Southampton, SO17 1BJ, U.K., email: wl@maths.soton.ac.uk

(Received September 8, 1999; revised June 12, 2000)

Abstract.
The central limit theorem for a normalized summation of random
number of i.i.d. random variables is well known. In this paper we
improve the central limit theorem by providing a two-term
expansion for the distribution when the random number is the first time that a simple
random walk exceeds a given level. Some numerical evidences are provided to show that this
expansion is more accurate than the simple normality approximation for a specific problem considered.

Key words and phrases:
Central limit theorem, expansion of a tail probability,
martingale, renewal theory, sequential analysis, stopping time, Wald's lemma.

**Source**
(TeX , DVI )