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

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.

References in corpus (1)

• Random weighted averages, partition structures and generalized arcsine laws