The average run length (ARL) is a key variable for assessing the properties of process control procedures. For continuous sampling procedures that are based on finite weighted sums (such as the moving sum technique) closed form expressions of the ARL are not available in the literature. For normally distributed random variables, Lai gives bounds for the ARL. In this paper we derive a lower bound of the ARL that (1) does not depend on normality and (2) in many situations is much sharper than the one obtained by Lai. Our results also imply that Lai's upper bound deviates from the true value less than the number of terms in the sum. Furthermore, we show that the applicability of Lai's bounds is not restricted to normally distributed control variables.
"Improved Bounds for the Average Run Length of Control Charts Based on Finite Weighted Sums." Ann. Statist. 18 (4) 1895 - 1899, December, 1990. https://doi.org/10.1214/aos/1176347887