Exact Convergence Rates of Functional Modulus of Continuity of a Wiener Process

Abstract:

      Let {W(t),t ≥ 0} be a standard Wiener process and S be the set of Strassen'sfunctions. In this paper we investigate the exact rates of convergence to zero of thevariables sup_0≤t≤1-h inf_f∈S sup_0≤x≤1 |(W(t+hx)-W(t))(2h log h^-1)^-1/2 - f(x)| and inf_0≤t≤1-h sup_0≤x≤1 |(W(t + hx) - W(t))(2hlogh^-1)^-1/2- f(x)| for any f ∈ S. As aconsequence, a relation between the modulus of non-differentiability and the fiunctionalmodulus of continuity for a Wiener process is established.