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And n=n0+l ^ ' — J we let 2 + 7fli- Whence, by induction, we have defined our Wiener process at every (x, r), 0^x<~>, where r is a non-negative dyadic rational number. 1. s. L = S . ) . The thus defined process obviously satisfies conditions (i)-(iii). In the rest of this Section we intend to prove that our process also satisfies (iv). 1. 1. 1) P l s u p sup \W(x9y+s)-W(x9y)\^vh1'2}^Ch-1e V* 2+£ holds for every positive v and 0

Our present definition of L gives that L+2 = L(k)+2 - j - i ^ log bTka^\ From here on this proof continues along the lines of that of £<1 above. Step 3. 5 are satisfied. s. i L bT 1 = T > M + ^bT if if e < 1 be the largest integer ' g = l. -,. }. 7. p=T be non-decreasing functions of T and define 52tT= ( 2 a r ( l o g ^ + l o g ( l o g j / ^ . 71 Wiener and some Related Gaussian Processes Further let L2tT=L2tT(aT9 b(}\ tip) (resp. L*2T=L*2 T(aT9 b%\ bf))be the set of rectangles R=[xl9 x2]X[yl9 y2]aD2tT for which X(R)^aT (resp.

2 . Since card Z j ( ? 15) ^ T Q ^ " ' ) P{ sup \W(R)\ ^ xa%*+4 j j 1 ' 2 } ^ 4cardL£(tf)1). s. 20). Step 2. 5 are satisfied.