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]}{X(t);t∈[0,1]}, can be modeled as a process with independent increments, satisfying the following limiting condition.liminfh↓0Ef(h-1/2[X(s+h)-X(s)])⩾Ef(X(1))almost surely for all 0⩽s<10⩽s<1, where Ef(X(1))<∞Ef(X(1))<∞ and f:R→Rf:R→R is a symmetric, continuous, convex function with f(0)=0f(0)=0, strictly increasing on R+R+ and satisfying the following growth condition:f(Kx)⩽Kpf(x),for a certainp∈[1,2), all K⩾K0 and all x>0(for example, f(x)=xp[A+Bln(1+Cx)]f(x)=xp[A+Bln(1+Cx)], with x>0x>0, p∈[1,2)p∈[1,2), A>0A>0 and B,C⩾0B,C⩾0).