A SIMPLE TIGHTNESS CONDITION FOR RANDOM ELEMENTS ON $ C([0,1]^2) $

Abstract

§ 1. Based on moments of variations of random elements on CV, 112), a simple sufficient condition for tightness** is considered. § 2. Let C2 C([0, 112) be the set of all real valued continuous functions on [0, 112 with the uniform topology. W. J. Park [2], [3] considered random elements on C2 to prove the existence of Wiener measure and invariance principle on C2. W. J. Park's sufficient condition for tightness (lemma 3 in [3]) is not necessarily easy to apply directly. The object of this paper is to give a handy sufficient condition for tightness of random elements on C2. We consider random elements {Z,i(t, s), t, s 1} n 1 satisfying :