ON THE MOMENTS OF THE MAXIMUM OF PARTIAL SUMS OF A FINITE NUMBER OF INDEPENDENT NORMAL VARIATES

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Abstract
The paper is concerned with the maximum Un of partial sums X1, X1+X2,…, X1+X2+…+Xn of n independent standard normal variates. The distribution of Un is of interest in the theory of storage. Suppose we have a reservoir of infinite capacity, which receives every year a random input, from rivers, etc., whose distribution is normal (μ, 1), and releases, for civil purposes, the mean discharge μ. The probability that, starting with an initial water level x, the reservoir will not run dry in the following n years is given by the distribution function Fn(x) of Un. The first and second moments of the variate Un have been previously obtained (Anis & Lloyd, 1953; Anis, 1955). Each of these moments was studied on its own merits, and no systematic method of attack was seen at that time. In the present paper, a method for obtaining all the moments is discussed. A recurrence relation is obtained which makes po their numerical evaluation.

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