AISM 54, 114-124
© 2002 ISM
(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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