AISM 54, 114-124
© 2002 ISM

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

Nan Wang1 and Wei Liu2

1Department of Social Statistics, University of Southampton, Southampton, SO17 1BJ, U.K., e-mail:
2Department of Statistics, University of Central Florida, Orlando, FL, U.S.A. and Department of Mathematics,University of Southampton, Southampton, SO17 1BJ, U.K., email:

(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.

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