Negatively reinforced balanced urn schemes

Abstract

We consider weighted negatively reinforced urn schemes with finitely many colours. An urn scheme is called negatively reinforced, if the selection probability for a colour is proportional to the weight w of the colour proportion, where w is a non-increasing function. Under certain assumptions on the replacement matrix R and weight function w, such as, w is differentiable and w(0) < ∞, we obtain almost sure convergence of the random configuration of the urn model. In particular, we show that if R is doubly stochastic the random configuration of the urn converges to the uniform vector, and asymptotic normality holds, if the number of colours in the urn are sufficiently large.

Positive reinforced generalized time-dependent Pólya urns via stochastic approximation

“…Finally they prove in the same paper that for d = 2, f convex and H irreducible, the limiting proportion converges towards the equilibrium points of the corresponding mean-field function resulting from the stochastic approximation approach. A concave feedback function, which includes the negative reinforcement regime, tends to equalize the asymptotic distribution of the proportion of different colors, whereas a convex f which includes the positive reinforcement regime tends to amplify the effect of the generating matrix H. In [19], the author proves CLT type results for the proportion vector of colors around the uniform distribution in the negative reinforcement setting, when H is double-stochastic and f Lipschitz.…”

“…where B n,i ∼ Bin (σ n , Ψ(θ n,i )) given F n , where Ψ was defined in (19). We can express the system of equations for i ∈ {1, ..., d} by…”