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Could anyone give me some references for the convergence rate of Markov chain arising from the random iteration of Lipschitz functions which moves as follows:

$$ X_{n+1}= f_{\omega_n}(X_n)$$ where $f_1,\dots, f_s$ are Lipschitz functions (with Lipschitz constants $L_i$ and $\sum_{k=1}^{s} p_kL_k <1$ )on $\mathcal X$, a metric space with a metric $d$ it is complete and separable say. $\omega_0,\omega_1,\dots, \omega_n,\dots$ are i.i.d discrete random variable taking values in $\{1,2,\dots,s<\infty\}$. $p_k=\mathbb P(\omega_i=k)$.

I found Diaconis and Freedman's Paper where they didn't talk about the finite number of function though and convergence rate is exponential in Prohorov's metric. I am looking for some reference on Wasserstein metric or so. Thanks for helping!

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It is much more natural and convenient to metrize the weak topology on the space of measures with the transportation (aka Kantorovich-Rubinshtein, aka 1-Wasserstein) metric, especially in what concerns iterated function systems and various convergence issues in this context, see Kaimanovich (1985).

Let me denote by $d^*$ the transportation metric on the space of probability measures on a metric state space $(X,d)$. If $f$ is a map of $X$ to itself with the Lipschitz constant $L$, then obviously $$ d^*(f(\lambda),f(\mu))\le L d^*(\lambda,\mu) \;. $$ Therefore if one has a family of such maps contracting on average, i.e., in the discrete case $$ A := \sum p_i \log L_i < 0 \;, $$ where $p_i$ are the weights of individual transformations $f_i$, and $L_i$ are their Lipschitz constants, then by the law of large numbers the time $n$ distribution of your Markov chain converges exponentially fast to its unique stationary distribution. Note that the right condition is $\sum p_i \log L_i<0$ rather than $\sum p_i L_i < 1$.

Moreover, for almost every Bernoulli sequence $f_{\omega_1},f_{\omega_2},\dots$ and any initial point $x\in X$ the sequence of iterations $f_{\omega_1} \dots f_{\omega_n}(x)$ exponentially fast converges to a point $z_{\omega_1,\omega_2,\dots}$ independent of $x$, and the distribution of these limit points coincides with the stationary distribution of the chain.

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  • $\begingroup$ The metric $d^*$ is defined on the space of measures, therefore whenever I talk about $d^*$ its arguments are measures. What I have said is essentially contained in Theorem 5.1 from Diaconis-Freedman. However, the fact that they were not aware of the transportation metric and argued in terms of the quite unnatural Prokhorov metric instead made their arguments way much longer than necessary. $\endgroup$ – R W Sep 1 at 14:11
  • $\begingroup$ No - $f(\mu)$ is the image of the measure $\mu$ under the map $f$. $\endgroup$ – R W Sep 1 at 15:25
  • $\begingroup$ Yes - it's also called the push forward measure. $\endgroup$ – R W Sep 1 at 16:37
  • $\begingroup$ The Lipschitz constant of the iterated random product $f_{\omega_1} \dots f_{\omega_n}(x)$ will almost surely behave like $e^{nA}$, where $A=\sum p_i \log L_i$ is negative. All this is in a pretty detailed way explained in Diaconis and Freedman, one just has to apply it to the transportation metric instead of Prokhorov's one. $\endgroup$ – R W Sep 2 at 21:44
  • $\begingroup$ They look to me like ad hoc moment conditions - they are of course satisfied when one deals with finitely many transformations and are only needed to make sure everything remains under control even if the distribution is spread out. $\endgroup$ – R W Sep 4 at 21:11

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