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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

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.
Full Text: DOI
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