Windings of Random Walks
1989; Institute of Mathematical Statistics; Volume: 17; Issue: 4 Linguagem: Inglês
10.1214/aop/1176991160
ISSN2168-894X
Autores Tópico(s)Point processes and geometric inequalities
ResumoLet $X_1, X_2, X_3, \ldots$ 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 $(1/2)\operatorname{sech}(\pi w/2)$.
Referência(s)