Consider the following nonlinear $\mathbb{R}$-valued stochastic recursive sequence:

$ X_{n+1} = F(X_n) + W_{n+1}, \quad (W_n)_{n\ge1} \stackrel{ \scriptsize \mathrm{i.i.d.} }{ \sim } \phi. $

How can we find explicitly the stationary distribution of $X_n$? Or at least some moments of it?

Edited: In my case $ F = X_n - \Phi(\alpha X_n) $ (where $\alpha$ is a constant and $\Phi$ is CDF of the standard normal distribution) and $W_n$ has Bernoulli distribution with success probability $p$.

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    $\begingroup$ It should be $F(X_n)$ (capital $x$), yes? Take $F={\rm id}$ to see that in general no stationary distribution exists. Do you have any other information on $F$? $\endgroup$ – UwF Mar 17 '14 at 15:25
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    $\begingroup$ Right: you'd like $F(x)-x\ll 0$ for large $x$ and $F(x)-x\gg 0$ for large negative $x$ to have a chance of having a stationary distribution. $F(x)=x/2$ would be fine. Then you could write down an explicit distribution. $\endgroup$ – Anthony Quas Mar 17 '14 at 15:57
  • $\begingroup$ UwF: Thanks for editing! In my case the stationary distribution probably exists, added information to the post. $\endgroup$ – user47855 Mar 20 '14 at 13:22
  • $\begingroup$ Anthony Quas: Yes, but the question really is: how to write down the explicit distribution provided its existence? $\endgroup$ – user47855 Mar 20 '14 at 13:29

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