Consider the following 
$f\_{t+1}(z)=p\_{12} f\_{t}(z/A)+ p\_{21} f\_{t}(z/B)+p\_{22} f\_{t}(z/(A+B))$, where $A$, $B$, and the $p$'s are constants and $f\_t$ is a probability distribution. Are there any nice distribution families that are preserved under the transformation? Fail that, are there $f\_t$ such that $f\_{t+1}$ has a closed form?


It's motivated by the following problem: 


Let there be two simple bonds that either default or pay off a return on investment (That may or may not be correlated), denote the bonds as random variables $Z\_1$ and $Z\_2$.


Now throw in a population of investors, with wealth following a distribution $W$, investing some fixed percentage of their income in the two bonds(investing a fixed percentage is a Nash equilibrium under the model I'm working with). The resulting after-investment wealth distribution will be a mixture of dilations of the original distribution, I'm trying to find a distribution to work with that will make things simple when studying the behavior of the system over time. 


Any ideas?



Finding a distribution family that is preserved under mixture.
One can rewrite the problem in terms of products of i.i.d. random variables as follows. 


Assume that $X\_t$ has distribution density $f\_t$. Then, the relation between $f\_t$ and $f\_{t+1}$ means that one can choose $X\_{t+1}=X\_tZ\_{t+1}$, where the $Z\_t$ are i.i.d. and $Z\_t=A$ or $B$ or $A+B$, with probabilities $p\_{12}A$, $p\_{21}B$ and $p\_{22}(A+B)$, respectively.


Hence, for the relation between $f\_t$ and $f\_{t+1}$ to make sense, one must assume that the three nonnegative numbers $p\_{12}A$, $p\_{21}B$ and $p\_{22}(A+B)$ sum to $1$, and when this is so, $X\_t=X\_0Z\_1Z\_2\cdots Z\_t$.


This tells you that:


* $E(X\_t)=E(X\_0)m^t$ for every $t$, with $m=E(Z\_1)$, that is, 
$$
m=p\_{12}A^2+p\_{21}B^2+p\_{22}(A+B)^2.
$$
* $t^{-1}\log X\_t$ converges almost surely to $\mu=E(\log Z\_1)$, that is, 
$$
\mu=p\_{12}A\log A+p\_{21}B\log B+p\_{22}(A+B)\log(A+B).
$$
* $\log X\_t$ follows the multinomial distribution with parameters $t$ and $(p\_{12}A,p\_{21}B,p\_{22}(A+B))$, or, more precisely, the convolution of this multinomial with the distribution of $\log X\_0$.


Unfortunately, these remarks do not help much if one is interested in closed form formulas. Sorry.


