# Iteration of the $\alpha$ parameter in a beta distribution yields a beta distribution?

I'd like to know whether starting from a beta distribution and iterating it in the way described below, I will get a stationary (beta) distribution again.

More specifically this is the problem I am facing:

Starting with a $\mathrm{Beta}(2,1.5)$, for each element $\phi$ drawn from this pdf, draw from an associated distribution; $\mathrm{Beta}(\phi+1,1.5)$. Iterating $n$ times, as $n$ goes to $\infty$, with this procedure do we get back a stationary distribution? Is it a beta distribution?

-
Numerical experiments (a sample of one million iterated $n=5000$ times) and some theoretical work indicate the answers are yes, the distribution stabilizes; but they suggest that no, it is not a Beta distribution. It takes a fairly large sample to distinguish it from a Beta(3/2,3/2) distribution: the stable distribution shifts some of its weight to the left. It takes a larger sample to distinguish it from some Beta distribution such as Beta[1.472, 1.507]. –  whuber Aug 29 at 14:51