Some further results on the height of lattice paths
type: Item
instance of: scholarly article
title: Some further results on the height of lattice paths
zbMATH Open document ID: 0766.60090
DOI: 10.1016/0378-3758(93)90002-N
author: Walter Katzenbeisser
author: Wolfgang Panny
published in: Journal of Statistical Planning and Inference
publication date: 1993-05-16T00:00:00Z
review text: 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 \(n\to\infty\).
reviewed by: Klaus Schürger
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 168236
zbMATH Keywords: height of lattice paths
zbMATH Keywords: random walk
zbMATH Keywords: asymptotic behaviour
MaRDI profile type: Publication
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full work available at URL: 0378-3758(93)90002-n
OpenAlex ID: W2023261374
description: scientific article; zbMATH DE number 168236