The number of distinct adjacent pairs in geometrically distributed words

TL;DR

This paper investigates the expected number of distinct adjacent pairs in sequences of independent geometric random variables, providing exact counts for short sequences and asymptotic behavior for large sequences.

Contribution

It derives an explicit expression for the expected number of distinct adjacent pairs in geometric words and analyzes its asymptotic growth as sequence length increases.

Findings

Exact counts for short sequence lengths

Asymptotic expression for large n

Growth rate of distinct pairs as sequence length increases

Abstract

A sequence of geometric random variables of length $n$ is a sequence of $n$ independent and identically distributed geometric random variables ($\Gamma_1, \Gamma_2, \dots, \Gamma_n$) where $\mathbb{P}(\Gamma_j=i)=pq^{i-1}$ for $1~\leq~j~\leq~n$ with $p+q=1.$ We study the number of distinct adjacent two letter patterns in such sequences. Initially we directly count the number of distinct pairs in words of short length. Because of the rapid growth of the number of word patterns we change our approach to this problem by obtaining an expression for the expected number of distinct pairs in words of length $n$. We also obtain the asymptotics for the expected number as $n \to \infty$.