Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part i

Satoshi Ishiwata, Hiroshi Kawabi, Ryuya Namba

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

In the present paper, we study central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a point of view of discrete geometric analysis developed by Kotani and Sunada. We establish a semigroup CLT for a non-symmetric random walk on a nilpotent covering graph. Realizing the nilpotent covering graph into a nilpotent Lie group through a discrete harmonic map, we give a geometric characterization of the limit semigroup on the nilpotent Lie group. More precisely, we show that the limit semigroup is generated by the sub-Laplacian with a non-trivial drift on the nilpotent Lie group equipped with the Albanese metric. The drift term arises from the non-symmetry of the random walk and it vanishes when the random walk is symmetric. Furthermore, by imposing the “centered condition”, we establish a functional CLT (i.e., Donsker-type invariance principle) in a Hölder space over the nilpotent Lie group. The functional CLT is extended to the case where the realization is not necessarily harmonic. We also obtain an explicit representation of the limiting diffusion process on the nilpotent Lie group and discuss a relation with rough path theory. Finally, we give an example of random walks on nilpotent covering graphs with explicit computations.

Original languageEnglish
Article number86
Pages (from-to)1-46
Number of pages46
JournalElectronic Journal of Probability
Volume25
DOIs
Publication statusPublished - 2020

Keywords

  • Albanese metric
  • Central limit theorem
  • Discrete geometric analysis
  • Modified harmonic realization
  • Nilpotent covering graph
  • Non-symmetric random walk
  • Rough path theory

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Fingerprint Dive into the research topics of 'Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part i'. Together they form a unique fingerprint.

Cite this