2004/04/02 by Robin Pemantle · 1 citation
Mathematics · #math.PR
published as Prob. Theor. and Rel. Fields, 92, 117 - 136 (1990) · 29 pages
arxiv created 2004/04/02 · arxiv updated 2009/12/01
This paper considers a class of non-Markovian discrete-time random processes on a finite state space 1,...,d. The transition probabilities at each time are influenced by the number of times each state has been visited and by a fixed a priori likelihood matrix, R, which is real, symmetric and nonnegative. Let Si(n) keep track of the number of visits to state i up to time n, and form the fractional occupation vector, V(n), where vi(n)=Si(n)/(sumj=1d Sj(n)). It is shown that V(n) converges to a set of critical points for the quadratic form H with matrix R, and that under nondegeneracy conditions on R, there is a finite set of points such that with probability one, V(n)->p for some p in the set. There may be more than one p in this set for which P(V(n)->p)>0. On the other hand P(V(n)->p)=0 whenever p fails in a strong enough sense to be maximum for H.