On the number of runs for Bernoulli arrays

Abstract

We introduce and motivate the study of (n + 1) × r arrays X with Bernoulli entries X k,j and independently distributed rows. We study the distribution of S n = r j =1 n k=1 X k,j X k+1,j , which denotes the number of consecutive pairs of successes (or runs of length 2) when reading the array down the columns and across the rows. With the case r = 1 having been studied by several authors, and permitting some initial inferences for the general case r > 1, we examine various distributional properties and representations of S n for the case r = 2, and, using a more explicit analysis, the case of multinomial and identically distributed rows. Applications are also given in cases where the array X arises from a Pólya sampling scheme.