Asymptotics of the run lengths of two control charts

The CUSUM control chart proposed by Page is a widely used in monitoring the quality of manufacturing processes. The Shiryayev-Roberts (S-R) control chart due to Shiryayev (1963) and Roberts (1988) is one of its competitors, This paper is concerned with the distribution properties of the run lengths of these two control charts. In context of continuous time, we first give the expansions of the higher moments of these run lengths. Then, we show that the asymptotic distributions of these run lengths are either some exponential distributions, or the distribution of the suprema of a standard Brownian motion, or some normal distributions, according to whether the μ<δ/2,μ =δ/2 and μ>δ/2. Here δ is the reference value of the above charts. Some similar results are also obtained in the context of discrete time.