## Electronic Journal of Probability

### The Beurling Estimate for a Class of Random Walks

#### Abstract

An estimate of Beurling states that if $K$ is a curve from $0$ to the unit circle in the complex plane, then the probability that a Brownian motion starting at $-\varepsilon$ reaches the unit circle without hitting the curve is bounded above by $c \varepsilon^{1/2}$. This estimate is very useful in analysis of boundary behavior of conformal maps, especially for connected but rough boundaries. The corresponding estimate for simple random walk was first proved by Kesten. In this note we extend this estimate to random walks with zero mean, finite $(3+\delta)$-moment.

#### Article information

Source
Electron. J. Probab., Volume 9 (2004), paper no. 27, 846-861.

Dates
Accepted: 13 October 2004
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465229713

Digital Object Identifier
doi:10.1214/EJP.v9-228

Mathematical Reviews number (MathSciNet)
MR2110020

Zentralblatt MATH identifier
1063.60066

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F99: None of the above, but in this section

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

Lawler, Gregory; Limic, Vlada. The Beurling Estimate for a Class of Random Walks. Electron. J. Probab. 9 (2004), paper no. 27, 846--861. doi:10.1214/EJP.v9-228. https://projecteuclid.org/euclid.ejp/1465229713

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