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 
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/
@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},
     year = {2008},
     volume = {20},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2008_20_1_a2/}
}
TY  - JOUR
AU  - I. A. Kruglov
TI  - The Kloss convergence principle for products of random variables with values in a compact group and distributions determined by a Markov chain
JO  - Diskretnaya Matematika
PY  - 2008
SP  - 38
EP  - 51
VL  - 20
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/DM_2008_20_1_a2/
LA  - ru
ID  - DM_2008_20_1_a2
ER  - 
%0 Journal Article
%A I. A. Kruglov
%T The Kloss convergence principle for products of random variables with values in a compact group and distributions determined by a Markov chain
%J Diskretnaya Matematika
%D 2008
%P 38-51
%V 20
%N 1
%U http://geodesic.mathdoc.fr/item/DM_2008_20_1_a2/
%G ru
%F DM_2008_20_1_a2

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Bakhturin Yu. A., Osnovnye struktury sovremennoi algebry, Nauka, Moskva, 1990 | MR | Zbl

[2] Gantmakher F. R., Teoriya matrits, Nauka, Moskva, 1988 | MR | Zbl

[3] Grenander U., Veroyatnosti na algebraicheskikh strukturakh, Mir, Moskva, 1965 | MR | Zbl

[4] Gorchinskii Yu. N., Kruglov I. A., Kapitonov V. M., “Voprosy teorii raspredelenii na konechnykh gruppakh”, Trudy po diskretnoi matematike, 1, 1997, 85–112 | MR

[5] Kolmogorov A. N., “Lokalnaya predelnaya teorema dlya klassicheskikh tsepei Markova”, Izv. AN SSSR. Ser. matem., 13:4 (1949), 281–300 | MR | Zbl

[6] Kloss B. M., “O veroyatnostnykh raspredeleniyakh na bikompaktnykh topologicheskikh gruppakh”, Teoriya veroyatnostei i ee primeneniya, 4:3 (1959), 255–290 | MR

[7] Kruglov I. A., “Svyaz tsepei Markova na konechnykh prostykh polugruppakh s fundamentalnymi gruppami”, Diskretnaya matematika, 18:2 (2006), 48–54 | MR | Zbl

[8] Kertis Ch., Rainer I., Teoriya predstavlenii konechnykh grupp i assotsiativnykh algebr, Nauka, Moskva, 1964 | MR

[9] Maksimov V. M., “O kompozitsionno skhodyaschikhsya posledovatelnostyakh mer na kompaktnykh gruppakh”, Teoriya veroyatnostei i ee primeneniya, 16:1 (1971), 52–66 | MR | Zbl

[10] Pontryagin L. S., Nepreryvnye gruppy, Nauka, Moskva, 1973 | MR | Zbl

[11] Sarymsakov T. A., Osnovy teorii protsessov Markova, GITTL, Moskva, 1954 | MR

[12] Kheier Kh., Veroyatnostnye mery na lokalno-kompaktnykh gruppakh, Mir, Moskva, 1981 | MR

[13] Prehn U., “Gleichverteilungseigenschaften von Verteilungsgesetzen zufälliger Summen, deren Summanden einer verallgemeinerten Markowschen Abhängigkeit unterworfen sind”, Math. Nachr., 49:1 (1971), 27–40 | DOI | MR | Zbl