Donkey walk and Dirichlet distributions

Gérard Letac

Statistics & Probability Letters, 2002, vol. 57, issue 1, 17-22

Abstract: The donkey performs a random walk (Xn)n[greater-or-equal, slanted]0 inside a tetrahedron with vertices A1,...,Ad as follows. For r=1,...,d and t=0,1,..., at time dt+r the donkey moves from the point Xdt+r-1 to a point Xdt+r such that the barycentric coordinates of Xdt+r with respect to A1,...,Ar-1,Xdt+r-1,Ar+1,...,Ad have a Dirichlet distribution depending on r. When the parameters are properly chosen, we compute the stationary distributions of the d homogeneous Markov chains (Xdt+r)t[greater-or-equal, slanted]0. For instance, if Xdt+r is uniformly chosen in the tetrahedron with vertices A1,...,Ar-1,Xdt+r-1,Ar+1,...,Ad then the stationary distribution of (Xdt)t[greater-or-equal, slanted]0 is Dirichlet with parameters (d,d-1,...,1).

Keywords: Markov; chains; in; a; tetrahedron; Random; walk; on; stochastic; matrices (search for similar items in EconPapers)
Date: 2002
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