Stochastic formation of hierarchies

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Abstract
In this paper, N particles el, · ··, eN occupying positions in finite state spaces S1, · ··, SN, respectively, are considered along with an element of a finite space S N+1 of hierarchies which describe the organization of the particles into pairs, pairs of pairs, etc. A stochastic process {Xt: t 0} on a probability measure space (Ω, , P) is a hierarchic process if it is a right-continuous Markov jump process taking on values in the state space Sj and having the property that the (N + 1)th component of Xt can jump from a hierarchy to a successor or antecedent of that hierarchy. Asymptotic distributions of perturbed hierarchic processes, bilateral processes, and unilateral processes are determined in terms of an interaction function and the asymptotic distributions of the particles in the absence of any interaction.

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