# Approximations for the boundary crossing probabilities of moving sums of random variables

## TL;DR

This paper develops highly accurate approximation methods for boundary crossing probabilities of moving sums of i.i.d. normal variables, applicable to various scenarios including non-normal data and unequal weights, with practical implications for ARL estimation.

## Contribution

It introduces a novel approximation approach transforming a discrete problem into a continuous one, improving accuracy for small window lengths and non-normal data.

## Key findings

- Approximations are highly accurate even for small window lengths.
- Method extends well to non-normal and unequal weight scenarios.
- Provides simple, accurate formulas for average run length (ARL).

## Abstract

In this paper we study approximations for the boundary crossing probabilities of moving sums of i.i.d. normal r.v. We approximate a discrete time problem with a continuous time problem allowing us to apply established theory for stationary Gaussian processes. By then subsequently correcting approximations for discrete time, we show that the developed approximations are very accurate even for small window length. Also, they have high accuracy when the original r.v. are not exactly normal and when the weights in the moving window are not all equal. We then provide accurate and simple approximations for ARL, the average run length until crossing the boundary.