  In this paper we study approximations for boundary crossing probabilities for
the moving sums of i.i.d. normal random variables. We propose approximating a
discrete time problem with a continuous time problem allowing us to apply
developed theory for stationary Gaussian processes and to consider a number of
approximations (some well known and some not). We bring particular attention to
the strong performance of a newly developed approximation that corrects the use
of continuous time results in a discrete time setting. Results of extensive
numerical comparisons are reported. These results show that the developed
approximation is very accurate even for small window length.
