On the sojourn time of a random walk in a strip
Let \(S_{n}=X_1+\dots+X_{n}\), \(n\geq1\), be a random walk with i.i.d. increments satisfying Cramér type conditions. For \(a\leq b\), let \(T_{n}=T_{n}([a,b])=\sum_{k=1}^nI\{a\leq S_{k}\leq b\}\) denote the sojourn time of the random walk in \([a,b]\) on the time interval \(\{1,\dots,n\}\). This paper is a continuation of the author's work [Sib. Math. J. 51, No. 1, 119--127 (2010; Zbl 1202.60067)], and corresponds to the second stage toward the final goal of finding the probabilities \(P(T_{n}=k)\). The main result gives asymptotic representations for the transform \(f_{a,b}(s,u,\lambda)=\sum_{n\geq1}z^nE[u^{T_{n}}e^{\lambda S_{n}}]\), where \(|z|<1\), \(|uz|<1\) and \(\operatorname{Re}\lambda>0\). The one-sided cases \(]-\infty,b]\) and \([a,\infty[\) are also considered.
scientific article; zbMATH DE number 5867019