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](https://statweb.stanford.edu/~cgates/PERSI/papers/iterate.pdf)
where they didn't talk about the finite number of functions though and convergence rate is exponential in Prohorov's metric. 


While talking about the convergence rate of probability measures towards the invariant measure, we need to talk about some metric on the space of probability measures, which also a question, which metric to use in this case to find the rate. 
Thanks. 


$\mu\_n(A)=\mathbb P(X\_n\in A)$ i.e $X\_n\sim \mu\_n$, 


I am looking for some reference on the Wasserstein metric. 
Thanks for helping!



Markov chain and random iteration of functions
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)](https://mathscinet.ams.org/mathscinet-getitem?mr=816292).


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.


