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

Abstract

The paper is concerned with the maximum U n of partial sums X 1 , X 1 +X 2 ,…, X 1 +X 2 +…+X n of n independent standard normal variates. The distribution of U n 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 F n ( x ) of U n . The first and second moments of the variate U n 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.