Some Results on Convolutions and a Statistical Application

Classes of distributions, of both discrete and continuous type, are introduced for which the right tail of the distribution is nonincreasing. It is shown that these classes are closed under convolution, thus providing sufficient conditions for nonincreasing right tails to be preserved under convolution. A start is made on verifying a conjecture concerning the extension to the left of nondecreasing right tails under successive convolution. The results give properties of the distributions of random walks on the integers. A statistical application is the verification of a conjecture of Sobel and Huyett (1957) concerning the minimal probability of correct selection for the usual indifference zone procedure for selecting the Bernoulli population with the largest success probability.