On hitting the high level by a random walk with delay at the origin (Q1975813)

Let \(\xi_1,\xi_2,\dots\) be independent identically distributed random variables with a nonzero absolutely continuous component and satisfying the Cramér condition. Let \(W_n\) be a random walk with delay at the origin, i.e., \(W_n=\max(0,W_{n-1}+\xi_n)\), \(W_0=a\geq 0\). For \(a\geq 0\) and \(b>0\), put \(T_{a,b}=\min\{n\geq 1:W_n\geq a+b\}\). Asymptotic representations for the distribution of \(T_{a,b}\), as \(a+b\to\infty\), are suggested. The author considers the cases \({\mathbb E} \xi_1<0\) and \({\mathbb E} \xi_1=0\) separately. Two different conditions are imposed on the initial state \(a\): \(a=\text{const}\) and \(a\to\infty\).