Suppose we have the following urn process with d different colors:
- We start with some deterministic initial composition $U_0 \in \mathbb N_0^d$. We denote the composition at time $n$ by $U_n$.
- In each step, we draw one ball uniformly at random and denote the color by $i$. We put the ball back into the urn together with $R_{ij}$ balls of color $j$ for each $j$.
- So we have $U_{n+1} = U_n + R^TI_n$, where $I_{n+1, i} = 1$ if the drawn color in step $n+1$ is of color $i$ and $I_{n+1, i} = 0$ otherwise.
If we denote the number of balls in the urn at time $n$ by $T_n = \sum_{k = 1}^n U_{n, k}$, then $X_n := \frac{U_n}{T_n}$ is the proportions vector.
Further, suppose that $R$ is nonnegative (except maybe for the diagonal but completely nonnegative is also ok) and irreducible and that the row sums are all equal to some fixed $s > 0$.
My question is: is there a fairly simple way (which mainly uses martingale techniques) to show that the proportions vector $X_n$ converges almost surely? I'm not interested in the (random) limit, I just want to be able to show that the proportions vector converges.
Also any source covering this or a technique to prove this would be appreciated.
I was trying to use this paper here, but I'm thinking that his appendix lemma doesn't work out which kind of destroys the whole proof.