Electronic Journal of Probability

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

Satoshi Ishiwata, Hiroshi Kawabi, and Ryuya Namba

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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.

Article information

Electron. J. Probab., Volume 25 (2020), paper no. 86, 46 pp.

Received: 25 October 2019
Accepted: 23 June 2020
First available in Project Euclid: 22 July 2020

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Mathematical Reviews number (MathSciNet)

Primary: 60F17: Functional limit theorems; invariance principles 60G50: Sums of independent random variables; random walks 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 22E25: Nilpotent and solvable Lie groups

central limit theorem non-symmetric random walk nilpotent covering graph discrete geometric analysis modified harmonic realization Albanese metric rough path theory

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Ishiwata, Satoshi; Kawabi, Hiroshi; Namba, Ryuya. Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part I. Electron. J. Probab. 25 (2020), paper no. 86, 46 pp. doi:10.1214/20-EJP486. https://projecteuclid.org/euclid.ejp/1595404963

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