Open Access
February 2005 Large deviations for template matching between point processes
Zhiyi Chi
Ann. Appl. Probab. 15(1A): 153-174 (February 2005). DOI: 10.1214/105051604000000576


We study the asymptotics related to the following matching criteria for two independent realizations of point processes XX and YY. 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 yt to the template over all yY∩[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 $\sqrt{l}$, 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.


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Zhiyi Chi. "Large deviations for template matching between point processes." Ann. Appl. Probab. 15 (1A) 153 - 174, February 2005.


Published: February 2005
First available in Project Euclid: 28 January 2005

zbMATH: 1068.60035
MathSciNet: MR2115040
Digital Object Identifier: 10.1214/105051604000000576

Primary: 60F10
Secondary: 60G55

Keywords: central limit theorem , large deviations , Point processes , template matching , Waiting times

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.15 • No. 1A • February 2005
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