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Learning models with continuous time parameter and multivariate point processes

Published online by Cambridge University Press:  14 July 2016

Helmut Pruscha
Helmut Pruscha*
Affiliation:
University of Munich
*
∗ Postal address: University of Munich, Mathematical Institute, Theresienstr. 39, D-8000 München 2, West Germany.
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Abstract

The concept of a learning model (or random system with complete connections) with continuous time parameter is introduced on the basis of the notion of a multivariate point process possessing an intensity. The stepwise transition probabilities in terms of the intensity are derived and a Monte Carlo method for simulating a sample is presented. By modelling the intensity process various types of learning models can be built. We propose a linear learning model which comprises the continuous-time Markov process as well as Hawkes's mutually exciting point process. We study the asymptotic behaviour of this linear model in terms of explosion or extinction and of convergence of some estimates. We close with some numerical results from computer simulations.

Keywords

RANDOM SYSTEM WITH COMPLETE CONNECTIONSINTENSITY PROCESSSIMULATION

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 20 , Issue 4 , December 1983 , pp. 884 - 890
Copyright
Copyright © Applied Probability Trust 1983 

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References

Brémaud, P. (1981) Point Processes and Queues. Springer-Verlag, New York.Google Scholar
Hawkes, A. G. (1971) Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.Google Scholar
Iosifescu, M. (1968) Processus aléatoires à liaisons complètes purement discontinus. C.R. Acad. Sci. Paris A 266, 1159–1161.Google Scholar
Iosifescu, M. and Theodorescu, R. (1969) Random Processes and Learning. Springer-Verlag, New York.CrossRefGoogle Scholar
Jacobsen, M. (1982) Statistical Analysis of Counting Processes. Lecture Notes in Statistics 12, Springer-Verlag, New York.Google Scholar
Jacod, J. (1975) Multivariate point processes. Z. Wahrscheinlichkeitsth. 31, 235–253.Google Scholar
Norman, M. F. (1972) Markov Processes and Learning Models. Academic Press, New York.Google Scholar
Ozaki, T. (1979) Maximum likelihood estimation of Hawkes' self-exciting point processes. Ann. Inst. Statist. Math. B 31, 145–155.Google Scholar
Pruscha, H. and Maurus, M. (1979) Analysis of primate communication by means of a multiresponse linear learning model. Rev. Roum. Math. Pures Appl. 24, 1371–1383.Google Scholar
Pruscha, H. and Theodorescu, R. (1981) On a non-markovian model with linear transition rule. Coll. Math. 44, 165–173.Google Scholar