EXPANSIONS FOR THE DISTRIBUTIONS OF SOME NORMALIZED SUMMATIONS OF RANDOM NUMBERS OF I.I.D. RANDOM VARIABLES

NAN WANG^{1} AND WEI LIU^{2}

1Department 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.