Laws relating runs and steps in gambler’s ruin

Let Xj denote a fair gambler’s ruin process on Z∩[−N,N] started from X0=0, and denote by RN the number of runs of the absolute value, |Xj|, until the last visit j=LN by Xj to 0. Then, as N→∞, N−2RN converges in distribution to a density with Laplace transform: tanh(λ)/λ. In law, we find: 2(limN→∞N−2RN)=limN→∞N−2LN. Denote by RN′ and LN′ the number of runs and steps respectively in the meander portion of the gambler’s ruin process. Then, N−1(2RN′−LN′) converges in law as N→∞ to the density (π/4)sech2(πx/2),−∞<x<∞.