Optimal sequential kernel detection for dependent processes

In many applications one is interested to detect certain (known) patterns in the mean of a process with the smallest delay. Using an asymptotic framework which allows to capture that feature, we study a class of appropriate sequential nonparametric kernel procedures under local nonparametric alternatives. We prove a new theorem on the convergence of the normed delay of the associated sequential detection procedure which holds for dependent time series under a weak mixing condition. The result suggests a simple procedure to select a kernel from a finite set of candidate kernels, and therefore may also be of interest from a practical point of view. Further, we provide two new theorems about the existence and an explicit representation of optimal kernels minimizing the asymptotic normed delay. The results are illustrated by some examples.