Windings of planar stable processes

TL;DR

This paper extends the understanding of the winding behavior of planar stable processes, providing new limit theorems for exit times from cones and establishing Laws of the Iterated Logarithm for the winding process.

Contribution

It introduces a new proof of an analogue of Spitzer's theorem for stable processes and derives limit theorems and LILs for the winding process of planar stable processes.

Findings

Limit theorems for exit times from cones of stable processes

Laws of the Iterated Logarithm for the winding process

New proof of Spitzer's theorem analogue for stable processes

Abstract

Using a generalization of the skew-product representation of planar Brownian motion and the analogue of Spitzer's celebrated asymptotic Theorem for stable processes due to Bertoin and Werner, for which we provide a new easy proof, we obtain some limit Theorems for the exit time from a cone of stable processes of index $\alpha\in(0,2)$. We also study the case $t\rightarrow0$ and we prove some Laws of the Iterated Logarithm (LIL) for the (well-defined) winding process associated to our planar stable process.