On Lerch's transcendent and the Gaussian random walk

A. J. E. M. Janssen, J. S. H. van Leeuwaarden

Let $X_1,X_2,...$ be independent variables, each having a normal distribution with negative mean $-\beta <0$ and variance 1. We consider the partial sums $S_n=X_1+...+X_n$, with $S_0=0$…