On the waiting time till some patterns occur in i.i.d. sequences

TL;DR

This paper provides general solutions for calculating the probabilities and expected waiting times for specific patterns to appear first in i.i.d. sequences, extending previous theoretical frameworks.

Contribution

It introduces new solutions to a system of linear equations to determine pattern occurrence probabilities and waiting times in i.i.d. sequences, building on Guibas and Odlyzko's work.

Findings

Derived probabilities for pattern first occurrence

Calculated expected waiting times for patterns

Analyzed conditional pattern occurrence scenarios

Abstract

In this paper we present some general solution of the system of linear equations formed by Guibas and Odlyzko in Th.3.3 \cite{Gui}. We derive probabilities for given patterns to be first to appear in random text and the expected waiting time till one of them is observed and also till one of them occurs given it is known which pattern appears first.