Waiting Time Distribution for the Emergence of Superpatterns

TL;DR

This paper analyzes the distribution of waiting times for the emergence of superpatterns in sequences of i.i.d. uniform random variables, focusing on the case where the alphabet size and pattern length are both three.

Contribution

It provides a detailed study of the waiting time distribution for superpatterns in the specific case of d=k=3, which was previously unexplored.

Findings

Derived the distribution of waiting times for superpatterns when d=k=3.

Identified the expected waiting time for the emergence of such superpatterns.

Provided insights into the probabilistic structure of pattern emergence in random sequences.

Abstract

Consider a sequence X_1, X_2,... of i.i.d. uniform random variables taking values in the alphabet set {1,2,...,d}. A k-superpattern is a realization of X_1,...,X_t that contains, as an embedded subsequence, each of the non-order-isomorphic subpatterns of length k. We focus on the non-trivial case of d=k=3 and study the waiting time distribution of tau=inf{t>=7: X_1,...,X_t is a superpattern}