Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-27T02:23:40.770Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinite-dimensional stochastic difference equations for particle systems and network flows

Published online by Cambridge University Press:  14 July 2016

R. W. R. Darling
R. W. R. Darling*
Affiliation:
University of South Florida
*
∗ Postal address: University of South Florida, Department of Mathematics, Tampa, FL 33620-5700, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let V be a countably infinite set, and let {Xn, n = 0, 1, ·· ·} be random vectors in which satisfy Xn = AnXn– 1 + ζn, for i.i.d. random matrices {An} and i.i.d. random vectors {ζ n}. Interpretation: site x in V is occupied by Xn(x) particles at time n; An describes random transport of existing particles, and ζ n(x) is the number of ‘births' at x. We give conditions for (1) convergence of the sequence {Xn} to equilibrium, and (2) a central limit theorem for n–1/2(X1 + · ·· + Xn), respectively. When the matrices {An} consist of 0's and 1's, these conditions are checked in two classes of examples: the ‘drip, stick and flow model' (a stochastic flow with births), and a neural network model.

Keywords

RANDOM MATRIXCENTRAL LIMIT THEOREMSTOCHASTIC FLOWINFINITE PARTICLE SYSTEMSTOCHASTIC AUTOMATA
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 325 - 344
Copyright
Copyright © Applied Probability Trust 1989 

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

Bougerol, Ph. (1987) Tightness of products of random matrices and stability of linear stochastic systems. Ann. Prob. 15, 40–74.CrossRefGoogle Scholar
Brandt, A. (1986) The stochastic equation Yn + 1 = AnYn + Bn with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google Scholar
Darling, R. W. R. and Mukherjea, A. (1988) Stochastic flows on a countable set J. Theoret. Prob. 1, 121–147.CrossRefGoogle Scholar
Dodziuk, J. and Kendall, W. S. (1985) Combinatorial Laplacians and isoperimetric inequality. In From Local Times to Global Geometry, etc., ed. Elworthy, K. D., Pitman Research Notes in Math 150, 68–74.Google Scholar
Hall, P. and Heyde, C. C. (1980) Martingale Limit Theory and its Application. Academic Press, New York.Google Scholar
Heyde, C. C. and Cohen, J. E. (1985) Confidence intervals for demographic projections based on products of random matrices. Theoret. Popn. Biol. 27, 120–153.CrossRefGoogle ScholarPubMed
Kesten, H. (1973) Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.CrossRefGoogle Scholar
Kesten, H. and Spitzer, F. (1984) Convergence in distribution of products of random matrices. Z. Wahrscheinlichkeitsch. 67, 363–386.CrossRefGoogle Scholar
Kipnis, C. (1987) Fluctuation des temps d'occupation d'un site dans l'exclusion simple symetrique. Ann. Inst. H. Poincaré 23, 21–36.Google Scholar
Liggett, T. M. (1985) Interacting Particle Systems. Springer-Verlag, New York.CrossRefGoogle Scholar
Mukherjea, A. (1987) Convergence in distribution of products of random matrices: a semigroup approach. Trans. Amer. Math. Soc. 303, 395–411.CrossRefGoogle Scholar
Petersen, K. (1983) Ergodic Theory. Cambridge University Press.CrossRefGoogle Scholar
Port, S. (1966) Equilibrium processes. Trans. Amer. Math. Soc. 124, 168–184.CrossRefGoogle Scholar
Schonmann, R. H. (1986) Central limit theorem for the contact process. Ann. Prob. 14, 1291–1295.CrossRefGoogle Scholar
Vervaat, W. (1979) On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.CrossRefGoogle Scholar