Abstract
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(θ), the waiting time until the matching score is above a given threshold θ. We show that whether the score is scalar- or vector-valued, (1/l)logWl(θ) 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(θ)=0}] by its rate function is established, and in the case where X is sufficiently mixing, the rates, after being centered and normalized by
Citation
Zhiyi Chi. "Large deviations for template matching between point processes." Ann. Appl. Probab. 15 (1A) 153 - 174, February 2005. https://doi.org/10.1214/105051604000000576
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