  Consider a generalized time-dependent P\'olya urn process defined as follows.
Let $d\in \mathbb{N}$ be the number of urns/colors. At each time $n$, we
distribute $\sigma_n$ 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 $\mathcal{R}$ assuming some monotonicity and
growth condition. The class $\mathcal{R}$ includes convex functions and the
classical case $f(x)=x^{\alpha}$, $\alpha>1$. 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 any more.
