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Download e-book for kindle: Gaussian Measures in Banach Spaces by Hui-Hsiung Kuo

By Hui-Hsiung Kuo

ISBN-10: 3540071733

ISBN-13: 9783540071730

ISBN-10: 3540375082

ISBN-13: 9783540375081

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Additional info for Gaussian Measures in Banach Spaces

Example text

J = 1,2,3 ..... introduced by Segal Using P r o p o s i t i o n n stands for the [41]. 2, we see that the om series of random variables 7. j=l n(e~) J J converges in 57 L2(~m) to a unique random variable, which we denote by n(h). (a) n(h) is normally distributed with mean 0 Corollary 4. i. and variance I hl 2 (b) Em[n(h)n(k)] = , h,k e H. if h l k then n(h) and n(k) are independent. Proof. (a) Em[e in(h)] = lim Em[e i N ~ = lim N§ Hence ~i n(ej) J= ] " h,ej >n(ej) E m [e i< ] 1 N 1 2 ~ e - 2 j=l = lim N +~ 1 N = lim e- 2 j~l 2 N +~ _ !

Following 2nk ~ n I < n 2 < intervals, [Sl, Sl + 1 i 1 2m--J]' [Sl + 2-~J' sl + -2mj-i - ....... so _ 1 2ml ... < n k- ' so ] 1 + 2mj ] ............ 41 and [So,S O + 1 ] , .... 9[So + ~ + ..... _- s2]. 2nl 2nl 2nk-i ' Let p = min (ml,n i) and q = max Then it is easy (mj,nk). to see that JX(Sl) - x(s2) I <~ 2a ~ p ( ~ ) ~ = 2a <. 3. Proof. w(i~,a,k,n ) -< 2 Note that I ,a,k, n is a cylinder set. x(t)-x(s) t-s, 1 _ a _la 2n(~- 2)e 1-2 . 2n(i-2~) Recall that is normally distributed with mean 0 and variance (t > S).

If I is a c y l i n d e r 1 t h e n w(I) < / ~ -- ~ a i _ 2 1 - 6 e - 89 set ' 1 6 = ~ - u. Remark. Proof. Note We use o n l y Hence, 1 lim a~ a that a rough 2 ~k a 2k(l-~) e- ~-~0 = 0 . 1_21_6e_89 estimate, < - - = namely, ~ k=o 2k(l-6)e- ~ k=0 (2 (_ a2 ~- ~2 ,(Y > 0). 1. generated Proof. w is o-additive in the o-field by R W~ n e e d of c y l i n d e r Let (Wiener) to s h o w t h a t sets w i t h empty In : In Ct~n~ t ~ ~x~ if I n is a d e c r e a s i n g intersection then sequence l i m w ( I n) = 0.

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Gaussian Measures in Banach Spaces by Hui-Hsiung Kuo

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