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

More specifically this is the problem I am facing:

Start with a Beta(2,1.5) for each element "phi" drawn from this pdf there is associated another distribution from which we draw: Beta("phi"+1,1.5). Iterating n times, as n goes to inf, this procedure do we get back a stationary distribution. Is it beta?

I think the problem could be rephrased also as follows, let the original starting point Beta be B0, we get a distribution (not sure if it would be beta) B2 associated with a probability measure CB2 that maps from every borel set A contained in [0,1] the following value:

CB2(A)=Integral(from0,1) CumulativeBeta(A|alpha="phi"+1,beta=2)dCumulativeB0(2,1.5)

and so on for all iterations.

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