Windings of Random Walks

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 $(1/2)\operatorname{sech}(\pi w/2)$.