Some first passage problems related to cusum procedures

Abstract

Let X 1, X 2,… be independent random variables, and set W n = max(0, W n-1 + X n ), W 0 = 0, n ⩾ 1. The so-called cusum (cumulative sum) procedure uses the first passage time T( h) = inf{ n ⩾ 1: W n ⩾ h}for detecting changes in the mean μ of the process. It is shown that lim h →∞ μET( h)/ h = 1 if μ > 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.