Deviation frequencies of Brownian path property approximations

TL;DR

This paper quantifies the robustness of classical Brownian motion path properties by analyzing deviation frequencies in typical almost sure approximations, covering key theorems and constructions in stochastic analysis.

Contribution

It introduces a systematic framework for measuring deviation frequencies of Brownian path properties in standard approximations, enhancing understanding of their robustness.

Findings

Quantified deviation frequencies for Lévys construction

Analyzed robustness of Kolmogorov continuity theorem

Examined deviation behaviors in laws of the iterated logarithm

Abstract

This case study proposes robustness quantifications of many classical sample path properties of Brownian motion in terms of the (mean) deviation frequencies along typical a.s.~approximations. This includes L\'evy's construction of Brownian motion, the Kolmogorov-Chentsov (and the Kolmogorov-Totoki) continuity theorem, L\'evy's modulus of continuity, the Paley-Wiener-Zygmund theorem, the a.s.~approximation of the quadratic variation as well as the laws of the iterated logarithm by Khinchin, Chung and Strassen, among others.