We discuss the convergence of the expected times until the partial sums of a sequence of independent, identically distributed random variables with zero means and unit variances first rise a height $h$ above their previous minimum as $h \rightarrow \infty$. We also consider the convergence as $r \rightarrow \infty$ of the expected times until the range of these partial sums exceeds a value $r$. Applications of these results to a quality control procedure and to queueing theory are mentioned.
"Convergence of Some Expected First Passage Times." Ann. Probab. 4 (6) 1027 - 1029, December, 1976. https://doi.org/10.1214/aop/1176995948