How to understand the stationary solution
of the stochatic equation: $X_{n+1}=A_n X_n+B_n$
And where can I find more information?
How to understand the stationary solution
of the stochatic equation: $X_{n+1}=A_n X_n+B_n$
And where can I find more information?
The solution is stationary in distribution, that is, one looks for a distribution $\mu$ such that, if $(A,B)$ is distributed like every $(A_n,B_n)$ and if $X$ is independent of $(A,B)$ and with distribution $\mu$, then the distribution of $AX+B$ is $\mu$ as well.
Equivalently, $\mu$ is a fixed point of the transformation of the space $\mathcal M_1^+$ of probability measures which sends any distribution of a random variable $X$ to the distribution of $AX+B$, with the independence assumptions explained above.
In the present case, $\mu$ should be the distribution of $ \sum_{k=1}^{+\infty}\left(\prod_{i=1}^{k-1}A_i\right)\cdot B_k, $ provided that this series converges almost surely (a condition which imposes that some hypotheses on the distribution of $A$ are met).
Section 2 of the paper Iterated Random Functions by Persi Diaconis and David Freedman is a classic on the subject.