# Nonlinear Markov process

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 $p$.

• 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 $F$? – UwF Mar 17 '14 at 15:25
• 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. – Anthony Quas Mar 17 '14 at 15:57
• UwF: Thanks for editing! In my case the stationary distribution probably exists, added information to the post. – user47855 Mar 20 '14 at 13:22
• Anthony Quas: Yes, but the question really is: how to write down the explicit distribution provided its existence? – user47855 Mar 20 '14 at 13:29