The Annals of Probability

Windings of Random Walks

Claude Belisle

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Abstract

Let $X_1, X_2, X_3, \ldots$ be a sequence of iid $\mathbb{R}^2$-valued bounded random variables with mean vector zero and covariance matrix identity. Let $S = (S_n; n \geq 0)$ be the random walk defined by $S_n = \sum^n_{i = 1} X_i$. Let $\phi(n)$ be the winding of $S$ at time $n$, that is, the total angle wound by $S$ around the origin up to time $n$. Under a mild regularity condition on the distribution of $X_1$, we show that $2\phi(n)/\log n \rightarrow_d W$ where $\rightarrow_d$ denotes convergence in distribution and where $W$ has density $(1/2)\operatorname{sech}(\pi w/2)$.

Article information

Source
Ann. Probab., Volume 17, Number 4 (1989), 1377-1402.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991160

Digital Object Identifier
doi:10.1214/aop/1176991160

Mathematical Reviews number (MathSciNet)
MR1048932

Zentralblatt MATH identifier
0693.60020

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60J65: Brownian motion [See also 58J65] 60F15: Strong theorems 60F17: Functional limit theorems; invariance principles

Keywords
Random walks Brownian motion windings weak and strong invariance principle asymptotic distributions

Citation

Belisle, Claude. Windings of Random Walks. Ann. Probab. 17 (1989), no. 4, 1377--1402. doi:10.1214/aop/1176991160. https://projecteuclid.org/euclid.aop/1176991160


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