fixed point distributional equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T03:41:50Z http://mathoverflow.net/feeds/question/113975 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/113975/fixed-point-distributional-equation fixed point distributional equation xn--qwertyuiop-86a 2012-11-20T18:43:44Z 2012-11-20T19:13:33Z <p>Hello. I'm trying to solve (or to get some information about) the following fixed point distributional equation $$X \overset{d}{=} \frac{z^2}{z^2+\sum_{i=1}^K X_i}$$ where $z$ is a deterministic positive number, $K$ is a non-negative integer random variable (say for example Poisson), $X_i$'s are i.i.d. copies of the random variable $X$. $K$ and $(X_i)_i$ are independent.</p> <p>In particular I'm interesed in proving existence and uniqueness of the solution and in computing $\mathbb{E}[X]$.</p> <p>Do you know some references to approach this problem? There already exists a general theory of fixed point distributional equations?</p>