2007/01/29 by Zach Dietz, Dietz, Zach, Sunder Sethuraman +1
Computer Science · Mathematics · #60F10 #60J10 #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60F10 #msc:60J10
paper · pdf · doi:10.48550/arxiv.math/0701798
24 pages, 2 figures
arxiv created 2007/01/29 · openalex publication_date 2007/01/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider finite-state time-nonhomogeneous Markov chains where the probability of moving from state i to state j≠ i at time n is G(i,j)/nζ for a ``generator'' matrix G and strength parameter ζ>0. In these chains, as time grows, the positions are less and less likely to change, and so form simple models of age-dependent time-reinforcing behaviors. These chains, however, exhibit some different, perhaps unexpected, asymptotic occupation laws depending on parameters. Although on the one hand it is shown that the asymptotic position converges to a point-mixture for all ζ>0, on the other hand, the average position, when variously 0<ζ<1, ζ>1 or ζ=1, is shown to converges to a constant, a point-mixture, or a distribution μG with no atoms and full support on a certain simplex respectively. The last type of limit can be seen as a sort of ``spreading'' between the cases 0<ζ<1 and ζ>1. In particular, when G is appropriately chosen, μG is a Dirichlet distribution with certain parameters, reminiscent of results in Polya urns.