  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 $(W_{t})_{t\in \left[ 0,T\right] }$ normalized by the global
modulus function, for all positive $\varepsilon $ 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.
