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.

On the number of runs for Bernoulli arrays

“…Proof. It follows directly from representation (12) and Theorem 1 that the probability function of S n in the context here of a Pólya urn is given by the above equations with B n,k,m = E[Z m a n,k (Z)], where Z This is verified by establishing that (i) Z = 4θ(1 − θ) ∼ Beta(2b/s − 1, 1 2 ) whenever θ ∼ Beta(b , b ), and (ii) by directly evaluating (14) via (15).…”

“…which implies that S n ∼ Bin(n, 1 2 ). For part (b), begin with standard operations to express α n as a polynomial in t. Write the λ 1 and λ 2 of Corollary 1 as λ 1 = (t + )/2 and λ 2 = (t − )/2, with = ρ + t 2 (1 − ρ), so that λ 1 − λ 2 = , and…”

“…(a) For p = 1 2 , we have S n ∼ Bin(n, 1 2 ). (b) Let ρ = 4p(1 − p), let n and k be nonnegative integers, and let a n,k (ρ…”

“…From both a combinatorial and probabilistic point of view, the probability distributions described in (9) and Theorem 1 form an interesting family on their own with parameter ρ, ρ ∈ (0, 1], with the case ρ = 1 corresponding to a Bin(n, 1 2 ) distribution. Moreover, as shown below, all of these distributions may be described by a large n normal approximation which extends the well-known result applicable to the Bin(n, 1 2 ) distribution.…”

“…We then proceed to more explicit representations (Corollary 1) for the particular case of multinomial rows (as in Example 1), which will lead to an explicit expression (Theorem 1) for the probability mass function of S n in the identically distributed case. The new family of distributions thus obtained, which is quite interesting on its own, may be viewed as one that contains (for fixed n) the Bin(n, 1 2 ) distribution and whose members admit a large n normal approximation (Theorem 2). Finally, although our setup does not encompass Pólya urn sampling schemes where the rows of X are dependent, we nevertheless derive by de Finetti's representation theorem novel implications for Pólya urns, as expanded upon in Section 3.…”