The Annals of Applied Probability

Large deviations for template matching between point processes

Zhiyi Chi

Full-text: Open access


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

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

First available in Project Euclid: 28 January 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Waiting times template matching large deviations point processes central limit theorem


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

Export citation


  • Abeles, M. and Gerstein, G. M. (1988). Detecting spatiotemporal firing patterns among simultaneously recorded single neurons. J. Neurophysiol. 60 909–924.
  • Chi, Z. (2001). Stochastic sub-additivity approach to conditional large deviation principle. Ann. Probab. 29 1303–1328.
  • Chi, Z., Rauske, P. L. and Margoliash, D. (2003). Pattern filtering for detection of neural activity, with examples from hvc activity during sleep in zebra finches. Neural Computation 15 2307–2337.
  • Comets, F. (1989). Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures. Probab. Theory Related Fields 80 407–432.
  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • Dayan, P. and Abbott, L. F. (2001). Theoretical Neuroscience. MIT Press.
  • Dembo, A. and Kontoyiannis, I. (1999). The asymptotics of waiting times between stationary processes, allowing distortion. Ann. Appl. Probab. 9 413–429.
  • Dembo, A. and Kontoyiannis, I. (2002). Source coding, large deviations, and approximate pattern matching. IEEE Trans. Inform. Theory 48 2276–2290.
  • Dembo, A. and Zeitouni, O. (1992). Large Deviations Techniques and Applications. Jones and Bartlett, Boston, MA.
  • Louie, K. and Wilson, M. A. (2001). Temporally structured replay of awake hippocampal ensemble activity during rapid eye movement sleep. Neuron 29 145–156.
  • Nádasdy, Z., Hirase, H., Czurkó, A., Csicsvari, J. and Buzsáki, G. (1999). Replay and time compression of recurring spike sequences in the hippocampus. J. Neurosci. 19 9497–9507.
  • Rio, E. (1995). The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann. Probab. 23 1188–1203.
  • Yang, E.-H. and Kieffer, J. C. (1998). On the performance of data compression algorithms based upon string matching. IEEE Trans. Inform. Theory 44 47–65.
  • Yang, E.-H. and Zhang, Z. (1999). On the redundancy of lossy source coding with abstract alphabets. IEEE Trans. Inform. Theory 45 1092–1110.