Electronic Communications in Probability

A radial invariance principle for non-homogeneous random walks

Nicholas Georgiou, Aleksandar Mijatović, and Andrew R. Wade

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Consider non-homogeneous zero-drift random walks in $\mathbb{R} ^d$, $d \geq 2$, with the asymptotic increment covariance matrix $\sigma ^2 (\mathbf{u} )$ satisfying $\mathbf{u} ^{\top } \sigma ^2 (\mathbf{u} ) \mathbf{u} = U$ and $\operatorname{tr} \sigma ^2 (\mathbf{u} ) = V$ in all in directions $\mathbf{u} \in \mathbb{S} ^{d-1}$ for some positive constants $U<V$. In this paper we establish weak convergence of the radial component of the walk to a Bessel process with dimension $V/U$. This can be viewed as an extension of an invariance principle of Lamperti.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 56, 11 pp.

Received: 25 August 2017
Accepted: 30 July 2018
First available in Project Euclid: 12 September 2018

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Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces 60F17: Functional limit theorems; invariance principles
Secondary: 60J60: Diffusion processes [See also 58J65]

non-homogeneous random walk invariance principle Bessel process

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Georgiou, Nicholas; Mijatović, Aleksandar; Wade, Andrew R. A radial invariance principle for non-homogeneous random walks. Electron. Commun. Probab. 23 (2018), paper no. 56, 11 pp. doi:10.1214/18-ECP159. https://projecteuclid.org/euclid.ecp/1536718009

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