On counts of Bernoulli strings and connections to rank orders and random permutations

A sequence of independent random variables {X1, X2, . . .} is called a B−harmonic Bernoulli sequence if P (Xi = 1) = 1 − P (Xi = 0) = 1/(i + B) i = 1, 2, . . ., with B ≥ 0. For k ≥ 1, the count variable Zk is the number of occurrences of the k-string (1, 0, . . . , 0 } {{ } k−1 , 1) in the Bernoulli sequence.. This paper gives the joint distribution PB of the count vector Z = (Z1, Z2, . . .) of strings of all lengths in a B−harmonic Bernoulli sequence. This distribution can be described as follows. There is random variable V with a Beta(B, 1) distribution, and given V = v, the conditional distribution of Z is that of independent Poissons with intensities (1− v), (1− v2)/2, (1− v3)/3, . . .. Around 1996, Persi Diaconis stated and proved that when B = 0, the distribution of Z1 is Poisson with intensity 1. Emery gave an alternative proof a few months later. For the case B = 0, it was also recognized that Z1, Z2, . . . , Zn are independent Poissons with intensities 1, 12 , . . . , 1 n . Proofs up until this time made use of hard combinational techniques. A few years later, Joffe et al, obtained the marginal distribution of Z1 as a Beta-Poisson mixture when B ≥ 0. Their proof recognizes an underlying inhomogeneous Markov chain and uses moment generating functions. In this note, we give a compact expression for the joint factorial moment of (Z1, . . . , ZN ) which leads to the joint distribution given above. One might feel that if Z1 is large, it will exhaust the number of 1’s in the Bernoulli sequence (X1, X2, . . .) and this in turn would favor smaller values for Z2 and introduce some negative dependence. We show that, on the contrary, the joint distribution of Z is positively associated or possesses the FKG property.