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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.
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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 

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