## The Annals of Applied Probability

### Large deviations for template matching between point processes

Zhiyi Chi

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 28 January 2005

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

#### 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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