## Electronic Communications in Probability

### A radial invariance principle for non-homogeneous random walks

#### Abstract

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

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

Dates
Accepted: 30 July 2018
First available in Project Euclid: 12 September 2018

https://projecteuclid.org/euclid.ecp/1536718009

Digital Object Identifier
doi:10.1214/18-ECP159

Mathematical Reviews number (MathSciNet)
MR3863912

Zentralblatt MATH identifier
1401.60133

#### Citation

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

#### References

• [1] Anshelevich, V.V., Khanin, K.M., and Sinai, Ya.G.: Symmetric random walks in random environments. Commun. Math. Phys. 85 (1982) 449–470.
• [2] Billingsley, P.: Convergence of Probability Measures, 2nd ed. John Wiley & Sons, Inc., New York, 1999.
• [3] Cherny, A.S.: On the strong and weak solutions of stochastic differential equations governing Bessel processes. Stochastics and Stochastics Reports 70 (2000) 213–219.
• [4] Ethier, S.N. and Kurtz, T.G.: Markov Processes. Characterization and Convergence. John Wiley & Sons, Inc., New York, 1986.
• [5] Georgiou, N., Menshikov, M.V., Mijatović, A., and Wade, A.R.: Anomalous recurrence properties of many-dimensional zero-drift random walks. Adv. in Appl. Probab. 48A (2016) 99–118.
• [6] Georgiou, N., Mijatović, A., and Wade, A.R.: Invariance principle for non-homogeneous random walks. arXiv:1801.07882
• [7] Gut, A.: Probability: A Graduate Course, 2nd ed. Springer, New York, 2013.
• [8] Lamperti, J.: A new class of probability limit theorems. J. Math. Mech. 11 (1962) 749–772.
• [9] Lawler, G.F.: Weak convergence of a random walk in a random environment. Commun. Math. Phys. 87 (1982) 81–87.
• [10] Menshikov, M., Popov, S., and Wade, A.: Non-homogeneous Random Walks. Cambridge University Press, Cambridge, 2016.
• [11] Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion, 3rd ed. Springer-Verlag, Berlin, 1999.
• [12] Zeitouni, O.: Random walks in random environment. Lectures on probability theory and statistics, pp. 189–312, Lecture Notes in Math., 1837, Springer, Berlin, 2004.