Vertex-reinforced Random Walk
نویسنده
چکیده
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)/( ∑d j=1 Sj(n)). It is shown that V(n) converges to 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.
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