Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-01T09:13:57.151Z Has data issue: false hasContentIssue false
  • English
  • Français

Selective interaction of a Poisson and renewal process: the spectrum of the intervals between responses

Published online by Cambridge University Press:  14 July 2016

A. J. Lawrance
A. J. Lawrance*
Affiliation:
IBM Thomas J. Watson Research Center, Yorktown Heights, New York
*
* On leave of absence from the University of Leicester, England during the year 1970–71.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with the spectrum of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b), (1971)). A generating function for all the pairwise joint distributions of the synchronous intervals following an average response is obtained and leads directly to the associated serial correlations. It is shown that these correlations are equivalent to those predicted on different assumptions by the general stationary point theory. The results are then used to obtain the interval spectrum, and to exhibit a relationship between the sum of the serial correlations and the variance-time function. Explicit results for the spectrum of the renewal inhibited Poisson process are given for gamma inhibitory distributions, and the qualitative behavior is determined. Possible further developments are briefly discussed.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 4 , December 1971 , pp. 731 - 744
Copyright
Copyright © Applied Probability Trust 1971 

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

Coleman, R. and Gastwirth, J. L. (1969) Some models for interaction of renewal processes related to neuron firing. J. Appl. Prob. 6, 38–58.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.CrossRefGoogle Scholar
Cox, D. R. and Lewis, P. A. W. (1970) Multivariate point processes. Proc. Sixth Berkeley Symp. Math. Statist. and Prob. (To appear).Google Scholar
Gilles, D. C. and Lewis, P. A. W. (1967) The spectrum of intervals of a geometric branching Poisson process. J. Appl. Prob. 4, 201–205.Google Scholar
Khintchine, A. Y. (1960) Mathematical Methods in the Theory of Queuing. Griffin, London.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
Lawrance, A. J. (1971) Selective interaction of a Poisson and renewal process: The dependency structure of the intervals between responses. J. Appl. Prob. 8, 170–183.Google Scholar
Leadbetter, M. R. (1966) On streams of events and mixtures of streams. J. R. Statist. Soc. B 28, 218–227.Google Scholar
Lewis, T. (1961) The intervals between regular events displaced in time by independent random deviations of large dispersion. J. R. Statist. Soc. B 23, 476–483.Google Scholar
Mcfadden, J. A. (1962) On the lengths of intervals in a stationary point process. J. R. Statist. Soc. B 24, 364–382.Google Scholar
Ten Hoopen, M. and Reuver, H. A. (1965) Selective interaction of two independent recurrent processes. J. Appl. Prob. 2, 286–292.Google Scholar
Ten Hoopen, M. and Reuver, H. A. (1967) Interaction between two independent recurrent time series. Information and Control 10, 149–158.Google Scholar