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Let us have interval $I = (i_1,i_2)$, function $f_1 : I \mapsto I$, function $f_2 : I \mapsto I$.

Let $x_0$, $x_1$, $x_2$, ... be series of random variables from interval $I$ denoting random walk. Let $x_0$ be deterministically midle of interval, $x_0 = \frac{i_1+i_2}{2}$. Then suppose that $x_{i+1}$ depends on $x_{i}$ randomly, either $x_{i+1} = f_1(x_i)$ or $x_{i+1} = f_2(x_i)$.

The question:

How do I compute stationary distribution of described random walk? i.e. what would be the distribution of $x_n$, after infinite number of steps, $n \rightarrow \infty$.

My attemt to solution (system of uncountably many linear equation):

Let $b(x)$ denote probability density of described stationary distribution. Let $p(x)$ denote the probability that the next step will be transformation $x$ to $f_1(x)$. Then probability of using $f_2$ is $(1-p(x))$.

Then the following condition must hold for stationary distribution $b(x)$.

$b(x) = p(f_1^{-1}(x)) b(f_1^{-1}(x)) + (1-p(f_2^{-1}(x))) b(f_2^{-1}(x))$.

I.e. probability of being in state $x$ equals to probability of being in previous states multiplied by probability state change.

$p$,$f_1$,$f_2$ are known, $b(x)$ is unknown for each $x \in I$ which leaves me with infinite number of equations with infinite number of variables. I am able to solve some simple recurent series, but I am unable to find solution to this recurrently specified function. (Other solution than constant zero).

I am looking for analytic solution. I am begging for your help.

My concrete functions: $f_1(x) = \frac{a+b x}{c + d x}$, $f_2(x) = \frac{a'+b' x}{c' + d' x}$.

As can be seen both $f_1$ and $f_2$ are shaped as hyperbola. Their domain restricts the hyperbola to nicely shaped continuous monotone curve. Futhermore suppose following: $x > f_1(x)$, $x < f_2(x)$, $lim_{x\rightarrow i_1} f_1(x) = i_1$, $lim_{x\rightarrow i_2} f_2(x) = i_2$.

Explanation: $f_2$ is a function transforming $x \in I$ to greater number up to $i_2$. Similarly $f_1$ transforms input to smaller numbers bounded by $i_1$.

$p(x) = 0.9 - 0.8 x$

$i_1 > 0$, $i_1+i_2 = 1$, $f_1(1-x)=1-f_2(x)$

Thanks for any help.

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  • $\begingroup$ I would also appreciate any suggestion on any tutorial how similar or related thinks are computed. (ie. stationary distributions on intervals or recurrently specified functions). I am not skilled mathematician and I never met anything similar. $\endgroup$
    – ghost guest
    Commented Feb 17, 2015 at 23:03
  • $\begingroup$ The question is more appropriate for Math.SE (as it is not a research-level question in mathematics). I would also suggest to fix the grammar and spelling to make the question easier to read. $\endgroup$
    – Boris Bukh
    Commented Feb 18, 2015 at 15:29

1 Answer 1

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What you are looking for is the stationary measure of an iterated function system. It is very unlikely that there is an analytic solution. Even in the case where $p(x)$ is constant and the $f_i$ are linear, this is not completely solved (you can look up this case under the name Bernoulli Convolution).

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