Renewal theory with a trend
Let \(Y, Y_1, \dots \) be i.i.d. random variables with finite mean 0 and set \(X_k=Y_k+k^\gamma \mu\) for \(k\geq 1\), and some \(\mu>0\) and \(0<\gamma<1\). Set \[ T_n=\sum_{k=1}^nY_k,\;S_n=\sum_{k=1}^nX_k,\quad n\geq 1. \] In the paper, some analogs of results from renewal theory for random walks (strong law, Marcinkiewich-Zygmund strong law, central limit theorem) are proved for the family of first passage times \[ \tau(t):=\min\{n: S_n>t\},\quad t\geq 0. \]
scientific article; zbMATH DE number 5932092