Rates of Convergence for Lévy’s Modulus of Continuity and Hinchin’s Law of the Iterated Logarithm

An Extreme-Value Analysis of the LIL for Brownian Motion

“…We can use our results to improve on the bounds of Dobric and Marano (2003) for the rate of convergence of E[sup s≥t B s (2sL 2 s) −1/2 ] to 1.…”

Lower Bounds for the Distribution of Suprema of Brownian Increments and Brownian Motion Normalized by the Corresponding Modulus Functions

“…Finally, we go beyond the results of the standard modulus of continuity by establishing a result which holds uniformly over δ. This last result allows us to establish rates of convergence; see [1].…”

“…The authors note that in their paper [1], the constants in Proposition 2.1 can be improved upon slightly by using the results of Theorem 5.…”

“…In section three, we exploit the Lévy-Ciesielski construction and the piecewise-linear truncated process (W n t ) t∈ [0,1] (the following section contains an explicit description of this process) over dyadic intervals to establish several results regarding the global modulus of continuity for Brownian motion.…”

“…Using a similar method, in section four, we establish the local modulus of continuity for the standard Brownian motion, with the same type of gains. That is, we construct functions l and q so that P sup t≤δ W t h (t) ≤ 1 + l(ε, δ) ≥ 1 − q(ε, δ) where h(x) = 2x ln ln 1 x , the local modulus of continuity. In all cases the Lévy-Ciesielski representation reveals how and why the logarithmic terms appear in the corresponding g and h functions.…”