Some further results on the height of lattice paths
Let \(X_ 1,X_ 2,\dots\) be i.i.d. random variables such that \(P(X_ 1=1)=P(X_ 1=-1)=1/2\). Put \(S_ m=X_ 1+\dots+X_ m\), \(m\geq 1\), \(S_ 0=0\), \(D^ +_ n=\max_{0\leq m\leq 2n}S_ m\), and let \(Q_ n\) denote the number of times the random walk \(S_ 0\), \(S_ 1,\dots,S_{2n}\) reaches its maximum. Let \(p_{n,r}(k)\) denote the probability (conditional on \(S_{2n}=0)\) of the event \(\{D^ +_ n=k\}\) given \(Q_ n=r\). Based on simple expressions for the joint distribution of \(D^ +_ n\) and \(Q_ n\) (conditional on \(S_{2n}=0)\) obtained by \textit{M. Dwass} [Ann. Math. Stat. 38, 1042-1053 (1967; Zbl 0162.502)] the authors obtain e.g. that, if \(r=O(n^{(1+\varepsilon)/2})\), \(\varepsilon>0\), \[ p_{n,r}(k)={2k+r-1\over n}\left(\exp\left[-{2k(k+r- 1)\over 2n-r+1}\right]+O(n^{-1+\varepsilon})\right)\quad\text{as } n\to\infty. \] They also obtain a result on the asymptotic behaviour of \(E[D_ n^{+s}\mid Q_ n=r]\) (conditional on \(S_{2n}=0)\) as \(n\to\infty\).
scientific article; zbMATH DE number 168236
scientific article