The weak stochastic realization problem for discrete-time counting processes

  • J. H. van Schuppen
Session 8 Identification And Detection
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 62)


The weak stochastic realization problem is considered for discrete-time stationary counting processes. Such processes take values in the countable infinite set N={0,1,2,...}. A stochastic realization is sought in the class of stochastic systems specified by a conditional distribution for the output given the state of Poisson type, and by a finite valued state process. In the paper a necessary and sufficient condition is derived for the existence of a stochastic realization in the above specified class.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    BERMAN, A. & R.J. PLEMMONS, Nonnegative matrices in the mathematical sciences, Academic Press, New York, 1979.Google Scholar
  2. [2]
    FAURRE, P., M. CLERGET & F. GERMAIN, Opérateurs rationnels positifs, Dunod, Paris, 1979.Google Scholar
  3. [3]
    LINDQUIST, A. & G. PICCI, On the stochastic realization problem, SIAM J. Control Optim., 17 (1979), pp. 365–389.Google Scholar
  4. [4]
    PICCI, G., On the internal structure of finite-state stochastic processes, in: Recent Developments in Variable Structure Systems, Economics, and Biology, Proc. of a U.S.-Italy Seminar, Taormina, Sicily, 1977, Lecture Notes in Econ. and Mathematical Systems, volume 162, Springer–Verlag, Berlin, 1978, pp. 288–304.Google Scholar
  5. [5]
    PICCI, G. & J.H. VAN SCHUPPEN, On the weak finite stochastic realization problem, Proc. Colloque ENST-CNET: Développements récents dans le filtrage et le contrôle des processes aléatoires, to appear; also report BW 184/83, Centre of Mathematics and Computer Science, Amsterdam, 1983.Google Scholar
  6. [6]
    RUCKEBUSCH, G., A state space approach to the stochastic realization problem, Proc. 1978 Int. Symp. on Circuits and Systems, New York, IEEE, New York, 1978, pp. 972–977.Google Scholar
  7. [7]
    SNYDER, D.L., Random point processes, J. Wiley & Sons, New York, 1975.Google Scholar
  8. [8]
    VAN SCHUPPEN, J.H., The strong finite stochastic realization problem-preliminary results, in: Analysis and Optimization of Systems, A. Bensoussan, J.L. Lions (eds.), Lecture Notes in Control nd Info. Sci., volume 44, Springer-Verlag, Berlin, 1982, pp. 179–190.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • J. H. van Schuppen
    • 1
  1. 1.Centre for Mathematics and Computer ScienceAmsterdamThe Netherlands

Personalised recommendations