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.
Categories: math.PR