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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 but it deals with only a special case of my problem.

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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, which provides an optimal transportation approach to some properties of such systems.

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  • $\begingroup$ Hi Benoit, thanks for the answer! I am more interested in the first case. Can you let me know how $X_t$ will behave then? I tried to work with expected values of $X_t$ and found that the expected value of $X_t$ is exponentially dependent on $t$. Is this correct? $\endgroup$
    – Shivin Srivastava
    Commented Sep 17, 2020 at 7:37
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    $\begingroup$ @ShivinSrivastava: as I wrote, $X_t=A_t X_1+B_t$ with $X_1, B_t>0$ and $A_t$ the exponential of a sum of iid rv. The usual limit theorem suffice to conclude an exponential growth (then depending on the precision you want, you may need to use various theorem and/or consider $B_t$). $\endgroup$
    – Benoît Kloeckner
    Commented Sep 17, 2020 at 13:55

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