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

Abstract

A sequence of independent random variables $\{X_1,X_2,\ldots\}$ is called a $B-$harmonic Bernoulli sequence if $P(X_i=1)=1-P(X_i=0) = 1/(i+B)\ i=1,2,\ldots$, with $B\ge 0$. For $k\ge 1$, the count variable $Z_k$ is the number of occurrences of the $k$-string $(1,\protect\underbrace{0,\ldots,0}_{k-1},1)$\vadjust{\vspace*{-2pt}} in the Bernoulli sequence\ldots\$. This paper gives the joint distribution $P_B$ of the count vector ${\bf Z} = (Z_1,Z_2,\ldots)$ 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 ${\bf Z}$ is that of independent Poissons with intensities $(1 -v),\ (1 - v^2)/2,\ (1-v^3)/3, \ldots$.

Around 1996, Persi Diaconis stated and proved that when $B=0$, the distribution of $Z_1$ is Poisson with intensity $1$. Emery gave an alternative proof a few months later. For the case $B=0$, it was also recognized that $Z_1,Z_2,\ldots,Z_n$ are independent Poissons with intensities $1, \frac{1}{2},\ldots, \frac{1}{n}$. Proofs up until this time made use of hard combinational techniques. A few years later, Joffe et al, obtained the marginal distribution of $Z_1$ as a Beta-Poisson mixture when $B\geq 0$. 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\break $(Z_1,\ldots,Z_N)$ which leads to the joint distribution given above. One might feel that if $Z_1$ is large, it will exhaust the number of $1$'s in the Bernoulli sequence $(X_1,X_2,\ldots)$ and this in turn would favor smaller values for $Z_2$ and introduce some negative dependence. We show that, on the contrary, the joint distribution of ${\bf Z}$ is positively associated or possesses the FKG property.