## The Annals of Applied Probability

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

### Large deviations for template matching between point processes

#### 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 *W*_{l}(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*)log*W*_{l}(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{*W*_{l}(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