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Nevertheless, these standards require some moment situation on the measure μ\mu (finite first moment for ray approximation, finite logarithmic moment for the strip approximation) to control the geometry of the random stroll. Later, Kaimanovich Kai00 prolonged this identification end result, between the Poisson boundary of the discrete group and the Furstenberg boundary of the ambient Lie group, to the extra basic case that μ\mu has finite entropy and finite logarithmic moment. We do not assume any second condition on the random walk and, in contrast with the previous rank eleven proof, we do not depend on pivoting concept.

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Let μ\mu be a very irreducible, bi-contracting probability measure on Γ\Gamma, with finite entropy. Now, we report the data of the random walk at time nn via the next process, that offers rise to 44 sets of partitions of the path space.. Let us additionally present that limit factors of the random walk are almost certainly pairs of transverse flags. Furthermore, when μ\mu has finite entropy, the equality holds if and provided that (X,λ)(X,\lambda) is the Poisson boundary. 2.1 is crucial to guarantee convergence of the random walk. The pin down partition is then constructed by replacing the space along a geodesic ray by the projection of the random walk onto the flats, utilizing the Cartan projection.}

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