Abstract
Consider a sequenceF 1,F 2,... of i.i.d. random transformations from a countable setV toV. Such a sequence describes a discrete-time stochastic flow onV, in which the position at timen of a particle that started at sitex isM n(x), whereM n =F n ∘F n−1 ∘⋯∘F 1. We give conditions on the law ofF 1 for the sequence (M n) to be tight, and describe the possible limiting law. an example called the block charge model is introduced. The results can be formulated as a statement about the convergence in distribution of products of infinite-dimensional random stochastic matrices. In practical terms, they describe the possible equilibria for random motions of systems of particles on a countable set, without births or deaths, where each site may be occupied by any number of particles, and all particles at a particular site move together.
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Darling, R.W.R., Mukherjea, A. Stochastic flows on a countable set: Convergence in distribution. J Theor Probab 1, 121–147 (1988). https://doi.org/10.1007/BF01046931
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DOI: https://doi.org/10.1007/BF01046931