Lower bounds for distribution of suprema of Brownian increments and Brownian motion normalized by the corresponding modulus functions

The L'evy-Ciesielski Construction of Brownian motion is used to determine non-asymptotic estimates for the maximal deviation of increments of a Brownian motion process $(W_t{)}_{t\in [0,T]}$ normalized by the global modulus function, for all positive $\epsilon$ and $\delta$. Additionally, uniform results over $\delta$ are obtained. Using the same method, non-asymptotic estimates for the distribution function for the standard Brownian motion normalized by its local modulus of continuity are obtained. Similar results for the truncated Brownian motion are provided and play a crucial role in establishing the results for the standard Brownian motion case.