Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T05:02:48.004Z Has data issue: false hasContentIssue false
  • English
  • Français

The rate and variance functions arising from the interaction of two stationary point processes

Published online by Cambridge University Press:  14 July 2016

A. J. Lawrance
A. J. Lawrance*
Affiliation:
University of Birmingham
*
∗Postal address: Department of Mathematical Statistics, University of Birmingham, P.O. Box 363, Birmingham B15 2TT, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper gives the mean-time and variance-time functions of responses arising from a general type of interaction between two stationary point processes which are independent of each other. The mean-time function is obtained exactly and explicitly, thereby much improving on earlier work. A Laplace-transformed version of the variance-time function is derived under the general assumptions, and new results for the asymptotic slope and intercept are obtained when one of the point processes is renewal. Attention is drawn to the difficulties of dealing with non-renewal and non-regenerative point processes.

Keywords

STATIONARY POINT PROCESSINTERACTION OF POINT PROCESSESRENEWAL THEORYVARIANCE-TIME FUNCTIONNEURON FIRING MODEL
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 287 - 296
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1966) Statistical Analysis of Series of Events. Methuen, London.Google Scholar
Fienberg, S. E. (1974) Stochastic models for single neuron firing trains: a survey. Biometrics 30, 399–427.Google Scholar
Lawrance, A. J. (1970a) Selective interaction of a stationary point process and a renewal process. J. Appl. Prob. 7, 483–489.Google Scholar
Lawrance, A. J. (1970b) Selective interaction of a Poisson and renewal process: first-order stationary point results. J. Appl. Prob. 7, 359–372.Google Scholar
Smith, W. L. (1954) Asymptotic renewal theorems. Proc. Roy. Soc. Edinburgh A64, 9–48.Google Scholar
Ten Hoopen, M. and Reuver, H. A. (1965) Selective interaction of two recurrent processes. J. Appl. Prob. 2, 286–292.Google Scholar
Ten Hoopen, M. and Reuver, H. A. (1968) Recurrent point processes with dependent interference with reference to neuronal spike trains. Math. Biosci. 2, 1–10.Google Scholar