Electronic Journal of Probability

Strong renewal theorems and local large deviations for multivariate random walks and renewals

Quentin Berger

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We study a random walk $\mathbf{S} _n$ on $\mathbb{Z} ^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbol{\alpha } =(\alpha _1,\ldots ,\alpha _d) \in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, i.e. a sharp asymptotic of the Green function $G(\mathbf{0} ,\mathbf{x} )$ as $\|\mathbf{x} \|\to +\infty $, along the “favorite direction or scaling”: (i) if $\sum _{i=1}^d \alpha _i^{-1} < 2$ (reminiscent of Garsia-Lamperti’s condition when $d=1$ [17]); (ii) if a certain local condition holds (reminiscent of Doney’s [13, Eq. (1.9)] when $d=1$). We also provide uniform bounds on the Green function $G(\mathbf{0} ,\mathbf{x} )$, sharpening estimates when $\mathbf{x} $ is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case $\alpha _i\equiv \alpha $, in the favorite scaling, and has even left aside the case $\alpha \in [1,2)$ with non-zero mean. Most of our estimates rely on new general (multivariate) local large deviations results, that were missing in the literature and that are of interest on their own.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 46, 47 pp.

Received: 3 August 2018
Accepted: 17 April 2019
First available in Project Euclid: 10 May 2019

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Primary: 60G50: Sums of independent random variables; random walks 60K05: Renewal theory 60F15: Strong theorems 60F10: Large deviations

multivariate random walks strong renewal theorems local large deviations

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Berger, Quentin. Strong renewal theorems and local large deviations for multivariate random walks and renewals. Electron. J. Probab. 24 (2019), paper no. 46, 47 pp. doi:10.1214/19-EJP308. https://projecteuclid.org/euclid.ejp/1557453645

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