The Annals of Applied Probability

Large deviations for template matching between point processes

Zhiyi Chi

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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 $\sqrt{l}$, 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.

Article information

Source
Ann. Appl. Probab., Volume 15, Number 1A (2005), 153-174.

Dates
First available in Project Euclid: 28 January 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1106922325

Digital Object Identifier
doi:10.1214/105051604000000576

Mathematical Reviews number (MathSciNet)
MR2115040

Zentralblatt MATH identifier
1068.60035

Subjects
Primary: 60F10: Large deviations
Secondary: 60G55: Point processes

Keywords
Waiting times template matching large deviations point processes central limit theorem

Citation

Chi, Zhiyi. Large deviations for template matching between point processes. Ann. Appl. Probab. 15 (2005), no. 1A, 153--174. doi:10.1214/105051604000000576. https://projecteuclid.org/euclid.aoap/1106922325


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