Central Limit Theorems for Non-Symmetric Random Walks on Nilpotent Covering Graphs: Part II

Satoshi Ishiwata, Hiroshi Kawabi, Ryuya Namba

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


In the present paper, as a continuation of our preceding paper (Ishiwata et al. 2018), we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a view point 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 Hölder 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.

Original languageEnglish
Pages (from-to)127-166
Number of pages40
JournalPotential Analysis
Issue number1
Publication statusPublished - 2021 Jun


  • Central limit theorem
  • Discrete geometric analysis
  • Nilpotent covering graph
  • Non-symmetric random walk
  • Rough path theory

ASJC Scopus subject areas

  • Analysis


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