Large deviations for template matching between point processes

We study the asymptotics related to the following matching criteria for two independent realizations of point processes X.X and Y.Y. Given l>0, X.[0,l) serves as a template. For each t>0, the matching score between the template and Y.[t,t+l) is a weighted sum of the Euclidean distances from y.t to the template over all y.Y.[t,t+l). The template matching criteria are used in neuroscience to detect neural activity with certain patterns. We first consider Wl(.), the waiting time until the matching score is above a given threshold .. We show that whether the score is scalar- or vector-valued, (1/l)logWl(.) converges almost surely to a constant whose explicit form is available, when X is a stationary ergodic process and Y is a homogeneous Poisson point process. Second, as l...., a strong approximation for .log[Pr{Wl(.)=0}] by its rate function is established, and in the case where X is sufficiently mixing, the rates, after being centered and normalized by .l, satisfy a central limit theorem and almost sure invariance principle. The explicit form of the variance of the normal distribution is given for the case where X is a homogeneous Poisson process as well.