An agent $A$ is performing a random walk on the number line. Let $X\_t$ be his position at time $t$. $X\_{t+1}$ is calculated according to the following rules:-


$ X\_{t+1} =$
\begin{cases}
 1 + X\_{t} & \text{with probability $a$} \\
 \alpha(1 + X\_{t}) & \text{with probability $b$} \\
 \beta(1+X\_t) & \text{with probability $c$}
 \end{cases}
such that $a+b+c=1$, $0<\alpha<1$, $1<\beta<\infty$ and $X\_0 = 0$.


What can we say about the statistics of $X\_t$?


I checked out Iterative Random Functions of [Diaconis and Freedman](http://finmath.stanford.edu/%7Ecgates/PERSI/papers/iterate.pdf) but it deals with only a special case of my problem.



A scaled random walk on the number line
It is known that:


* if $b\log \alpha+ c\log \beta>0$, then $X\_t\to\infty$ almost surely,
* if $b\log \alpha+ c\log \beta<0$, then $X\_t$ is recurrent and has a unique stationary measure, with (if $c>0$) tail $\mu([t,+\infty))\sim c t^\kappa$ for some $c>0$, $\kappa<0$. One can also derive a Central Limit Theorem, and many other limit theorems.


The first point is easily proven: setting $\phi\_0(x)=x+1$, $\phi\_1(x)=\alpha(x+1)$ and $\phi\_2(x)=\beta(x+1)$, you have
$$ X\_t = \phi\_{i\_t} \circ \cdots \circ \phi\_{i\_2}(X\_1) $$
and the composition $\phi\_{i\_t}\circ \cdots \circ \phi\_{i\_2}$ is an affine map of the form $x\mapsto A\_t x+B\_t$ where $A\_t, B\_t$ are random, with $B\_t>0$ and $\log A\_t\to\infty$ almost surely as the sum of i.i.d. random variable of positive expectation.


The second point is mostly handled by Diaconis & Freedman's paper and references cited therein: Theorem 1.1 applies and gives the unique stationary measure, while the tail estimate is given by Kesten and Goldie, mentioned in Section 2.4. For CLT and more, some references are given in the last paragraph of section 1.1 in [my own work on the topic](https://perso.math.u-pem.fr/kloeckner.benoit/papiers/StationaryIFS.pdf), which provides an optimal transportation approach to some properties of such systems.


