Application of Smirnov Words to Waiting Time Distributions of Runs

TL;DR

This paper analyzes the waiting times for specific run patterns in infinite i.i.d. random words, providing formulas for expectations and variances of first occurrences of runs of given lengths.

Contribution

It introduces a novel application of Smirnov words to derive explicit formulas for waiting times of runs in i.i.d. sequences.

Findings

Explicit formulas for expected waiting times of h-runs of individual letters

Variance calculations for waiting times of runs

Expected waiting times for h-runs of multiple distinct letters

Abstract

Consider infinite random words over a finite alphabet where the letters occur as an i.i.d. sequence according to some arbitrary distribution on the alphabet. The expectation and the variance of the waiting time for the first completed $h$-run of any letter (i.e., first occurrence of $h$ subsequential equal letters) is computed. The expected waiting time for the completion of $h$-runs of $j$ arbitrary distinct letters is also given.