First occurrence of a word among the elements of a finite dictionary in random sequences of letters

Emilio De Santis,Fabio Spizzichino

doi:10.1214/ejp.v17-1878

Emilio De Santis, Fabio Spizzichino

Open Access

https://doi.org/10.1214/ejp.v17-1878

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Abstract

In this paper we study a classical model concerning occurrence of words in a random sequence of letters from an alphabet. The problem can be studied as a game among $(m+1)$ words: the winning word in this game is the one that occurs first. We prove that the knowledge of the first $m$ words results in an advantage in the construction of the last word, as it has been shown in the literature for the cases $m=1$ and $m=2$ [CZ1,CZ2]. The last word can in fact be constructed so that its probability of winning is strictly larger than $1/(m+1)$. For the latter probability we will give an explicit lower bound. Our method is based on rather general probabilistic arguments that allow us to consider an arbitrary cardinality for the alphabet, an arbitrary value for $m$ and different mechanisms generating the random sequence of letters.

Highlights

The theme of the occurrence of words in random sequences of letters from an alphabet is a rather classical one in discrete probability
The related literature has a long tradition and papers with new insights and deep results continue to appear from time to time
Different types of interesting problems arise in this field and many important papers appeared in the related literature; see in particular [4, 1, 2, 5, 6, 7] and references cited therein

Summary

Introduction

The theme of the occurrence of words in random sequences of letters from an alphabet is a rather classical one in discrete probability. The related literature has a long tradition and papers with new insights and deep results continue to appear from time to time. This topic has, among others, the following interesting aspects: it has a number of important applications and it is characterized by surprising results which, at a first glance, can sometimes appear even contradictory. One interesting problem considers a finite set of given words, a dictionary, and concerns the probability that a fixed word occurs as the first.

Construction of efficient words and probability of winning

Discussion and final remarks