Some first passage problems related to cusum procedures

Let X1,X2,... be independent random variables, and set Wn = max(0,Wn-1 + Xn), W0 = 0, n [greater-or-equal, slanted] 1. The so-called cusum (cumulative sum) procedure uses the first passage time T(h) = inf{n [greater-or-equal, slanted] 1: Wn[greater-or-equal, slanted]h}for detecting changes in the mean [mu] of the process. It is shown that limh-->[infinity] [mu]ET(h)/h = 1 if [mu] > 0. Also, a cusum procedure for detecting changes in the normal mean is derived when the variance is unknown. An asymptotic approximation to the average run length is given.