Approximations and Inequalities for Moving Sums

In this article accurate approximations and inequalities are derived for the distribution, expected stopping time and variance of the stopping time associated with moving sums of independent and identically distributed continuous random variables. Numerical results for a scan statistic based on a sequence of moving sums are presented for a normal distribution model, for both known and unknown mean and variance. The new R algorithms for the multivariate normal and t distributions established by Genz et al. (2010) provide readily available numerical values of the bounds and approximations.