A Feyman-Kac approach to a paper of Chung and Feller on fluctuations in the coin-tossing game

TL;DR

This paper extends classical results on coin-tossing fluctuations by deriving new probability formulas for odd numbers of tosses using a Feynman-Kac approach, broadening the understanding of partial sum distributions.

Contribution

It introduces a novel application of the Feynman-Kac method to derive explicit probabilities for the number of positive partial sums in odd coin-toss sequences, generalizing previous even-toss results.

Findings

Derived explicit formulas for P(N_{2n+1}=r) for all r.

Extended classical results to odd numbers of coin tosses.

Applied Feynman-Kac methodology to a classical probability problem.

Abstract

A classical result of K. L. Chung and W. Feller deals with the partial sums $S_k$ arising in a fair coin-tossing game. If $N_n$ is the number of "positive" terms among $S_1, S_2,\dots,S_n$ then the quantity $P(N_{2n}=2r)$ takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for $P(N_{2n+1}=r)$, $r=0,1,2,\dots,2n+1$. We get to this result by adapting the Feynman-Kac methodology.