Stable windings at the origin

TL;DR

This paper investigates the behavior of windings of stable processes around the origin at both small and large times, using new techniques from self-similar Markov processes to extend previous results.

Contribution

It introduces novel methods to analyze windings at the origin for stable processes, covering both large and small time scales.

Findings

Characterization of windings at the origin for large times

Analysis of upcrossings at small times

Extension of previous planar results to one-dimensional setting

Abstract

In 1996, Bertoin and Werner [5] demonstrated a functional limit theorem, characterising the windings of pla- nar isotropic stable processes around the origin for large times, thereby complementing known results for planar Brownian mo- tion. The question of windings at small times can be handled us- ing scaling. Nonetheless we examine the case of windings at the the origin using new techniques from the theory of self-similar Markov processes. This allows us to understand upcrossings of (not necessarily symmetric) stable processes over the origin for large and small times in the one-dimensional setting.