pr.probability - Finding a distribution family that is preserved under mixture.

Below I propose a solution to the difference equation
$$ f_{t+1}(z) =p_{12} f_t(z/A) + p_{21}f_t(z/B) + p_{22} f_t(z/(A+B)),$$

where the $p_{ij}$'s are positive, $$p_{12}+p_{21}+p_{22}le 1$$ and $f_t$ is a pdf.

By integrating both sides of the given equation from $-infty$ to $infty$ we obtain after appropriate change of variables in the right hand side integrals

$$ 1=Ap_{12}+Bp_{21}+(A+B)p_{22}. $$

Now we look for a solution of our initial problem in the form

$$ f_t (z) =sum_{n=0}^{infty} q_n (t) z^n .$$

Substituting the above ansatz into our equation yields after elementary manipulation

$$ q_n (t+1) =left ( p_{12} A^{-n} +p_{21} B^{-n} +(A+B)^{-n} right ) q_n (t). $$

For fix $n$, the last equation is a linear difference equation that can be easily solved to produce
$$ a_n(t) = b_n t^{ p_{12} A^{-n} +p_{21} B^{-n} +(A+B)^{-n}}$$,
where $a_n$ is independent of $t$ i.e. it's a pure constant.
Finally we obtain the closed-form solution
$$ f_t (z) =sum_{n=0}^{infty} b_n t^{ p_{12} A^{-n} +p_{21} B^{-n} +(A+B)^{-n}} z^n.$$
Note that $$f_1(z) =sum_{n=0}^{infty} b_n z^n.$$
Thus our solution is completely specified given the initial pdf $f_1(z)$.
What is left is to tackle the issue of convergence and possible look for alternative representation of the solution.