On generalized renewal measures and certain first passage times (Q1201171)

The author considers generalized renewal functions \(U_ \varphi(t)=\sum_{n\geq 0}\varphi(n)P(S_ n\leq t)\) for arbitrary random walks \(S_ n\) with positive drift \(\mu\) and functions \(\varphi\) for which \(t^{-\alpha}\varphi(t)\) is slowly varying. The main result states that for \(\alpha\geq 0\), \(E((X^ -_ 1)^ 2\varphi(X_ 1))<\infty\) is equivalent to the finiteness of \(U_ \varphi(t)\) (for all or for some \(t)\), and also to \((t\varphi(t))^{-1}U_ \varphi(t)\sim(\alpha+1)^{- 1}\mu^{-\alpha-1}\), as \(t\to\infty\); if \(\alpha=0\), it must be assumed additionally that \(\varphi\) is ultimately increasing. The above moment condition still implies the other statements if \(\alpha\in(-1,0]\) and \(\varphi\) is ultimately decreasing. Similar results are given for the increments \(U_ \varphi(t+h)-U_ \varphi(t)\) and the sequence of maxima \(M_ n=\max_{0\leq j\leq n}S_ j\) instead of \(S_ n\).