Convergence rate for the longest T-contaminated runs of heads. Paper with detailed proofs

TL;DR

This paper analyzes the asymptotic behavior and convergence rates of the longest T-contaminated runs of heads in coin tossing, providing improved approximation results and detailed proofs.

Contribution

It introduces new asymptotic distributions and convergence rate improvements for the longest T-contaminated head runs, extending previous results.

Findings

Asymptotic distribution for first hitting time of T-contaminated runs for T=1,2

Limit theorem for the length of the longest T-contaminated run

Enhanced approximation rate for the distribution of the longest run

Abstract

We study the length of $T$-contaminated runs of heads in the well-known coin tossing experiment. A $T$-contaminated run of heads is a sequence of consecutive heads interrupted by $T$ tails. For $T=1$ and $T=2$ we find the asymptotic distribution for the first hitting time of the $T$ contaminated run of heads having length $m$; furthermore, we obtain a limit theorem for the length of the longest $T$-contaminated head run. We prove that the rate of the approximation of our accompanying distribution for the length of the longest $T$-contaminated head run is considerably better than the previous ones. For the proof we use a powerful lemma by Cs\'aki, F\"oldes and Koml\'os.