The (simple) Polya urn contains $a \in \mathbf N$ black and $b \in \mathbf N $ white balls at the initial time $t=0$, and, at each time $t \in \mathbf Z_{+}$, a ball is picked uniformly at random in the urn and put back in the urn together with a ball of the same color to give the composition of the urn at time $t+1$.
The Polya urn is very well understood : there is a.s. convergence of the proportion of black balls (as a bounded martingale) and the limit is $\beta(a,b)$ distributed.
An alternative description of the process is : if $(U_t)_{t \in \mathbf Z_{+}}$ is a sequence of independent uniform random variables on $[0,1]$, the number of blacks balls at time $t$ satisfies
$$X_0=a , \text{ and } X_{t+1}=X_{t}+ 1_\left\{ U_t < \frac{X_t}{t+a+b}\right\}$$
I know two methods to get the aforementioned limiting distribution of the proportion $X_t/(t+a+b)$ : computing explicitly the marginal of $X_t$ (see Durrett, Probability, Theory and Examples, section 4.3.2), using the combinatorics of the problem, or use an alternative construction of the process (the first edition of the book on Markov chains by Levin Peres Wilmer used this derivation) also based on exchangeability. These proofs are not very robust though.
My question : is it possible to give a proof based on the recursive equation in distribution displayed above (e.g. using convergence of the generating functions)?
