Summary
Consider a random walk of law μ on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG ∞. We study the relation between the spacesL ∞ (G ∞, ℬa,Q) andL ∞ (G ∞, ℬi,Q) where ℬa and ℬi stand for the asymptotic and invariant σ-algebras, respectively. We obtain a factorizationL ∞ (G ∞, ℬa,Q) ≊L ∞ (G ∞, ℬi,Q)⊗L ∞ (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powersμ n.
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Jaworski, W. On the asymptotic and invariantσ-algebras of random walks on locally compact groups. Probab. Th. Rel. Fields 101, 147–171 (1995). https://doi.org/10.1007/BF01375822
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DOI: https://doi.org/10.1007/BF01375822