Another approach to Brownian motion

Motivated by the central limit theorem for weakly dependent variables, we show that the Brownian motion { X ( t ) ; t ∈ [ 0 , 1 ] } , can be modeled as a process with independent increments, satisfying the following limiting condition. lim inf h ↓ 0 E f ( h - 1 / 2 [ X ( s + h ) - X ( s ) ] ) ⩾ E f ( X ( 1 ) ) almost surely for all 0 ⩽ s < 1 , where E f ( X ( 1 ) ) < ∞ and f : R → R is a symmetric, continuous, convex function with f ( 0 ) = 0 , strictly increasing on R + and satisfying the following growth condition: f ( Kx ) ⩽ K p f ( x ) , for a certain p ∈ [ 1 , 2 ) , all K ⩾ K 0 and all x > 0 (for example, f ( x ) = x p [ A + B ln ( 1 + Cx ) ] , with x > 0 , p ∈ [ 1 , 2 ) , A > 0 and B , C ⩾ 0 ).