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- W3049668712 abstract "In the present paper, as a continuation of our preceding paper [10], we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a viewpoint of discrete geometric analysis developed by Kotani and Sunada. We introduce a one-parameter family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We first prove a semigroup CLT for the family of random walks by realizing the nilpotent covering graph into a nilpotent Lie group via discrete harmonic maps. The limiting diffusion semigroup is generated by the homogenized sub-Laplacian with a constant drift of the asymptotic direction on the nilpotent Lie group, which is equipped with the Albanese metric associated with the symmetrized random walk. We next prove a functional CLT (i.e., Donsker-type invariance principle) in a Holder space over the nilpotent Lie group by combining the semigroup CLT, standard martingale techniques, and a novel pathwise argument inspired by rough path theory. Applying the corrector method, we finally extend these CLTs to the case where the realizations are not necessarily harmonic." @default.
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- W3049668712 date "2020-08-14" @default.
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- W3049668712 title "Central Limit Theorems for Non-Symmetric Random Walks on Nilpotent Covering Graphs: Part II" @default.
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- W3049668712 doi "https://doi.org/10.1007/s11118-020-09851-7" @default.
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