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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?

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Do you have any constraints on the $p$'s? – Steve Huntsman Mar 18 '10 at 15:06
Ah, sorry. They're basically mixture weights, they should be positive and sum to something less then or equal to 1. – David Shor Mar 18 '10 at 16:23
up vote 2 down vote accepted

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.

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One related concept is the p-stable distribution. It's a distribution such that the combination $\sum_i a_i X_i$, if $X_i$ are all i.i.d with respect to the distribution, is distributed as $\|a\|_p X$, where $X$ is also governed by the distribution.

Gaussians are 2-stable, and the Cauchy distribution is 1-stable. There is also a $(1/2)$-stable distribution. In general though, such distributions don't exist for $p > 2$.

Wikipedia has a useful entry on this.

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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.

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