Sample Module for the Wiener Process on a VC Class of R^d

Sample path behavior is studied for the Wiener processes W indexed by classes C of subsetsof the Borel δ-field of R^d. For 0＜a_n ≤n^d, a_n is nondecreasing and a_n/n^d is nonmcreasing, afunction Ψn is found in some cases such that limsup_n→∝sup｛|W（C^（n））|/Ψn:C^（n）∈C|C^（n）|≤a_n｝=1 a.s. This generalies the Csörgö-Révész’s increments results for a Wiener process.