A study of counts of Bernoulli strings via conditional Poisson processes

TL;DR

This paper introduces a new framework using conditional Poisson processes to analyze counts of specific patterns in Bernoulli sequences, providing a unified approach that extends previous results and includes many new sequences.

Contribution

It offers a novel characterization of the joint distribution of Bernoulli string counts via mixtures of Poisson measures, broadening the scope of analysis to new Bernoulli sequences.

Findings

Unified framework for Bernoulli string counts

Characterization as mixtures of Poisson measures

Includes all previously studied sequences and new ones

Abstract

We say that a string of length $d$ occurs, in a Bernoulli sequence, if a success is followed by exactly $(d-1)$ failures before the next success. The counts of such $d$-strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic $d$-cycle counts in random permutations. In this note, we give a new framework, in terms of conditional Poisson processes, which allows for a quick characterization of the joint distribution of the counts of all $d$-strings, in a general class of Bernoulli sequences, as certain mixtures of the product of Poisson measures. This general class includes all Bernoulli sequences considered before, as well many new sequences.