Runs in coin tossing: a general approach for deriving distributions for functionals

TL;DR

This paper introduces a unified method for deriving distributions of run-related functionals in sequences of independent coin tosses, generalizing previous approaches and providing new insights and asymptotic results.

Contribution

It presents a general identity and recursion for computing distributions of run functionals, unifying and extending existing methods.

Findings

Derived a general recursion for run distributions

Unified previous approaches to run analysis

Provided asymptotic results and posed open questions

Abstract

We take a fresh look at the classical problem of runs in a sequence of i.i.d.\ coin tosses and derive a general identity/recursion which can be used to compute (joint) distributions of functionals of run types. This generalizes and unifies already existing approaches. We give several examples, derive asymptotics, and pose some further questions.