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Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence. (French) Zbl 0558.60009
Let G be a connected semisimple Lie group, $$\mu$$ a probability measure on G, $$\{Y_ i\}^ a$$sequence of independent G-valued random variables with distribution $$\mu$$. The authors investigate the asymptotic behaviour of the random walk $$X_ n:=Y_ 1...Y_ n$$. The problem was discussed by different authors in the past, see e.g. H. Furstenberg [Trans. Am. Math. Soc. 108, 377-428 (1963; Zbl 0203.191)], H. Furstenberg and H. Kesten [Ann. Math. Stat. 31, 457-469 (1960; Zbl 0137.355)], V. N. Tutubalin [Teor. Veroyatn. Primen. 10, 19-32 (1965; Zbl 0147.171); English translation in Theory Probab. Appl. 10, 25-27 (1965)], A. Raugi [Bull. Soc. Math. France, Suppl., Mém. 54, 5-118 (1977; Zbl 0389.60003)].
There was additionally supposed that $$\mu$$ is absolutely continuous to the Haar measure or, at least, $$\mu$$ is spread out (”etalée”), or (G a group of matrices) that $$\mu$$ is concentrated on the subsemigroup of positive matrices. In the paper under review these conditions are not supposed to hold, it contains and precises therefore the preceding results.
Let $$\Delta$$ be the set of roots, $$G=NAK$$ an Iwasawa decomposition with $$A=\exp {\mathcal A}$$. For $$\alpha \in \Delta,\phi_{\alpha}: A\to {\mathbb{R}}_+$$ denotes the corresponding homomorphism. Let $$W\subseteq {\mathcal A}$$ be a Weyl chamber and $$G=K\cdot \exp \bar W\cdot K$$ a polar decomposition of G. Finally let $$\Gamma$$ be a maximal amenable subgroup of G and $$B:=G/\Gamma$$ the homogeneous space. Let $$\xi$$ : $$G\to B=G/\Gamma$$ be the canonical projection. A sequence $$\{g_ n\}\subseteq G$$ is called contracting if for the representation $$g_ n=x_ na_ nk_ n$$ in a polar decomposition holds: $$\phi_{\alpha}(a_ n)\to 0$$ for $$\alpha\in \Delta$$. The authors introduce further a notion of irreducibility for a probability on B and of total irreducibility for a subgroup of G. Let $$T_{\mu}$$ resp. $$G_{\mu}$$ be the closed subsemigroup resp. subgroup generated by the support of $$\mu$$. With this preparations the main result (Theorem 2.6) is:
Suppose that $$T_{\mu}$$ contains a contracting sequence and that $$G_{\mu}$$ is totally irreducible. Then there exists a unique $$\mu$$- invariant probability measure $$\nu$$ on B (which is irreducible). For every irreducible measure $$\lambda$$ on B, y,g$$\in G$$ the sequence of measures $$(y\cdot X_ n\cdot g\lambda)$$ converges a.s. to a $$\epsilon_{y\cdot Z}$$. Let further $$yX_ ng=x_ na_ nk_ n$$ $$(x_ n,k_ n\in K$$, $$a_ n\in \exp \bar W)$$ be the representation in the polar decomposition, then $$\xi (X_ n)\to y\cdot Z\quad and\quad \phi_{\alpha}(a_ n)\to 0\quad a.s.\quad for\quad \alpha \in \Delta.$$
In the sequel applications to the asymptotic behaviour of $$H(u,X_ n)$$ (where $$H: K\times G\to {\mathcal A}$$ is defined via the Iwasawa decomposition $$\kappa \cdot \gamma =n(\kappa,\gamma)\exp H(\kappa,\gamma)k(\kappa,\gamma))$$ are given. The hypothesis and the results are explained considering the example $$G=SL(d,{\mathbb{R}})$$. Moreover in § 4, 5 additional and more precise results for SL(d,$${\mathbb{R}})$$ are obtained. Especially in § 5 B almost sure convergence and a CLT-type- theorem for (functions of) the random walk $$X_ n$$ are proved under the additional hypotheses that $$\mu$$ is spread out and that some moment conditions hold.
Reviewer: W.Hazod

##### MSC:
 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60B10 Convergence of probability measures 60G50 Sums of independent random variables; random walks 22E46 Semisimple Lie groups and their representations 43A05 Measures on groups and semigroups, etc.
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##### References:
 [1] Azencott, R.: Espaces de Poisson des groupes localement compacts. Lecture Notes, no 148. Berlin-Heidelberg-New York: Springer 1970 · Zbl 0239.60008 [2] Borel, A.: Introduction aux groupes arithmétiques. Paris: Hermann 1969 · Zbl 0186.33202 [3] Bougerol, P.: Théorème central limite local sur certains groupes de Lie. (à paraître) · Zbl 0488.60013 [4] Conze, J.P., Guivarc’h, Y.: Remarques sur la distalité dans les espaces vectoriels. C.R.A.S. t. 278, p. 1083–1086, 1974 · Zbl 0275.54028 [5] Dunford, N., Schwartz, J.T.: Linear operators. New York: Interscience 1958 · Zbl 0084.10402 [6] Furstenberg, H.: A Poisson formula for semi-simple Lie groupe Ann. Math.77, 335–386 (1963) · Zbl 0192.12704 · doi:10.2307/1970220 [7] Furstenberg, H.: Non commuting random, products. Trans. Amer. Math. Soc.108, 377–428 (1963) · Zbl 0203.19102 · doi:10.1090/S0002-9947-1963-0163345-0 [8] Furstenberg, H.: Boundary theory and stochastic processes on homogeneous spaces. Proc. Symp. Pure Math.26, 193–229 (1972) [9] Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Statist.31, 457–469 (1960) · Zbl 0137.35501 · doi:10.1214/aoms/1177705909 [10] Guivarc’h, Y.: Quelques propriétés asymptotiques des produits de matrices aléatoires. Lecture notes 774. Berlin-Heidelberg-New York: Springer 1980 [11] Guivarc’h, Y.: Sur les exposants de Liapunoff des marches aléatoires à pas Markovien. Séminaire de Rennes 1981 [12] Guivarc’h, Y.: Croissance polynominale et périodes des fonctions harmoniques. Bulletin SMF101, 333–379 (1973) [13] Helgason, S.: Differential geometry and symetric spaces. Academic Press: New York 1962. · Zbl 0111.18101 [14] Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966 · Zbl 0148.12601 [15] Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. A.M.S. Trans.26, 200–325 (1950) [16] Le Page, E.: Théorèmes limites pour les produits de matrices. Séminaire de probabilités de Rennes 1981; note du, C.R.A.S. t. 290 p. 559–662, 1980 [17] Mostow, G.D.: Strong rigidity of locally symmetric spaces. Ann. Math. Studies, Princeton: University Press 1973 · Zbl 0265.53039 [18] Moore, C.C.: Amenable subgroups of semi-simple groups and proximal flows. Israel J. Math.34, 121–138 (1979) · Zbl 0431.22014 · doi:10.1007/BF02761829 [19] Norman, F.: Markov processes and learning models. New York: Academic Press, vol. 84, 1972 · Zbl 0262.92003 [20] Oseledec, V.I.: A multiplicative ergodic theorem. Trans. Moscow Math. Soc.19, 197–231 (1968) [21] Raghunathan, M.S.: A proof of Oseledec multiplicative theorem. Israel J. Math.32, 356–362 (1979) · Zbl 0415.28013 · doi:10.1007/BF02760464 [22] Raugi, A.: Pèriodes des fonctions harmoniques bornées. Séminaire de probabilités de Rennes, 1978 [23] Raugi, A.: Fonctions harmoniques et théorèmes limites pour les marches gléatoires sur les groupes. Bull. Soc. Math. France, mémoire 54, 1977 [24] Tutubalin, V.N.: Some theorems of the type of the strong law of large numbers. Theory Probability Appl.14, 313–319 (1969) · Zbl 0196.20904 · doi:10.1137/1114039 [25] Tutubalin, V.N.: On limit theorems for a product of random matrices. Theory Probability Appl.10, 25–27 (1965) · Zbl 0147.17105 · doi:10.1137/1110002 [26] Virtser, A.D.: Central limit theorem for semi-simple Lie groups Theory Probability Appl.15, 667–687 (1970) · Zbl 0232.60004 · doi:10.1137/1115074
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