Author:
Ruszel Wioletta M.,Thacker Debleena
Abstract
AbstractConsider a generalized time-dependent Pólya urn process defined as follows. Let $$d\in \mathbb {N}$$
d
∈
N
be the number of urns/colors. At each time n, we distribute $$\sigma _n$$
σ
n
balls randomly to the d urns, proportionally to f, where f is a valid reinforcement function. We consider a general class of positive reinforcement functions $$\mathcal {R}$$
R
assuming some monotonicity and growth condition. The class $$\mathcal {R}$$
R
includes convex functions and the classical case $$f(x)=x^{\alpha }$$
f
(
x
)
=
x
α
, $$\alpha >1$$
α
>
1
. The novelty of the paper lies in extending stochastic approximation techniques to the d-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls anymore.
Publisher
Springer Science and Business Media LLC