Limiting behaviors for longest consecutive switches in an IID Bernoulli sequence

TL;DR

This paper establishes precise bounds for the maximum length of consecutive switches in IID Bernoulli sequences, extending previous work on head-runs and unbiased coin-tossing, with potential applications across various fields.

Contribution

It provides sharp lower and upper bounds for the longest consecutive switches, generalizing earlier results and broadening understanding of pattern extremes in Bernoulli sequences.

Findings

Derived sharp bounds for longest consecutive switches

Extended classical results to more general Bernoulli sequences

Potential applications in multiple scientific fields

Abstract

In this paper we mainly discuss sharp lower and upper bounds for the length of longest consecutive switches in IID Bernoulli sequences. This work is an extension of results in Erd\H{o}s and R\'{e}v\'{e}sz (1975) for longest head-run and Hao et al. (2021) for longest consecutive switches in unbiased coin-tossing, and might be applied to reliability theory, biology, quality control, pattern recognition, finance, etc.