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We study independent and identically distributed random iterations ofcontinuous maps defined on a connected closed subset $S$ of the Euclidean space$\mathbb{R}^{k}$. We assume the maps are monotone (with respect to a suitablepartial order) and a "topological" condition on the maps. Then, we prove theexistence of a pullback random attractor whose distribution is the uniquestationary measure of the random iteration, and we obtain the synchronizationof random orbits. As a consequence of the synchronization phenomenon, afunctional central limit theorem is established.

Random iterations of maps on $\mathbb{R}^{k}$: asymptotic stability, synchronization and functional central limit theorem