A simple proof of the formula of Solov'ev--Nielsen--Blom for the expected waiting time

TL;DR

This paper provides a straightforward combinatorial proof of the well-known formula for the expected waiting time until a specific pattern appears in random data, simplifying previous complex proofs.

Contribution

It introduces a simple combinatorial proof of the Solov'ev--Nielsen--Blom formula, making the result more accessible and easier to understand.

Findings

Simplified proof of the expected waiting time formula

Clarification of combinatorial aspects of pattern occurrence

Enhanced understanding of pattern waiting time calculations

Abstract

Solov'ev (1966), Nielsen (1973), and Blom (1982) independently showed a formula for the expected waiting time until a given finite pattern first occurs in random data. In this paper, we give a simple and combinatorial proof of the formula.