Geodesic
The Kloss convergence principle for products of random variables with values in a~compact group and distributions determined by a~Markov chain
Diskretnaya Matematika, Tome 20 (2008) no. 1, pp. 38-51

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In this paper we study the weak convergence of distributions for products of random variables with values in a compact group provided that the distributions of the factors are defined by a finite simple homogeneous irreducible Markov chain. We show that after an appropriate shift the sequence of distributions of these products converges weakly to the normalised Haar measure on some closed subgroup of the initial group, in other words, the convergence principle due to B. M. Kloss holds true, which has been established earlier for products of independent factors. We describe conditions on the Markov chain and on the initial distributions which guarantee that the limit behaviour of the distribution of the products is similar to the limit behaviour of the distributions of some products of independent random variables. 
@article{DM_2008_20_1_a2,
     author = {I. A. Kruglov},
     title = {The {Kloss} convergence principle for products of random variables with values in a~compact group and distributions determined by {a~Markov} chain},
     journal = {Diskretnaya Matematika},
     pages = {38--51},
     publisher = {mathdoc},
     volume = {20},
     number = {1},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2008_20_1_a2/}
}
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I. A. Kruglov. The Kloss convergence principle for products of random variables with values in a~compact group and distributions determined by a~Markov chain. Diskretnaya Matematika, Tome 20 (2008) no. 1, pp. 38-51. http://geodesic.mathdoc.fr/item/DM_2008_20_1_a2/