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VOL. 45 | 2004 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.

## Information

Published: 1 January 2004
First available in Project Euclid: 28 November 2007

zbMATH: 1268.60011
MathSciNet: MR2126893

Digital Object Identifier: 10.1214/lnms/1196285386

Subjects:
Primary: 60C35
Secondary: 60E05

Keywords: Bernoulli , cycles , factorial moments , Random permutations , rank order statistics , records , Strings