Gaussian Processes, Moving Averages and Quick Detection Problems

Abstract

In this paper, we are interested in moving averages of the type $\int^t_0 f(t - s) dX(s)$, where $X(t)$ is a Wiener process and $\int^\infty_0 f^2(t) dt < \infty$. By a suitable choice of the weighting function $f$, such processes can be used to detect a change in the drift of $X(t)$. First passage times of these moving-average processes and more general Gaussian processes are studied. Limit theorems for Gaussian processes and Gaussian sequences which include these moving-average processes and their discrete-time analogs as special cases are also proved.