A new method is used to study the optimal stopping set corrected for discreteness introduced by Chernoff and studied by Chernoff and Petkau. The discrete boundary is asymptotically the optimal boundary for a Wiener process translated downward by a constant amount. This amount is shown to be an "excess over the boundary" term, and this method yields it as a simple integral involving the characteristic function of the random walk.
"Comments on a Problem of Chernoff and Petkau." Ann. Probab. 14 (3) 1058 - 1063, July, 1986. https://doi.org/10.1214/aop/1176992458