The Contraction Method on C([0,1]) and Donsker's ... - Eurandom
The Contraction Method on C([0,1]) and Donsker's ... - Eurandom
The Contraction Method on C([0,1]) and Donsker's ... - Eurandom
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(Bt) t<br />
Decompositi<strong>on</strong> of Brownian Moti<strong>on</strong><br />
d<br />
=<br />
<br />
1 1<br />
√2 1{t≤1/2}B2t + 1 {t>1/2}B1 + √21 {t>1/2}B ′ <br />
2t−1<br />
t<br />
= 1<br />
√ 2 ϕ2(B) + 1<br />
√ 2 ψ2(B ′ ), (3)<br />
with (c<strong>on</strong>tinuous <strong>and</strong> linear) functi<strong>on</strong>s ϕβ, ψβ : C([0, 1]) → C([0, 1])<br />
ϕβ(f )(t) = 1 {t≤1/β}f (βt) + 1 {t>1/β}f (1),<br />
ψβ(f )(t) = 1 {t≤1/β}f (0) + 1 {t>1/β}f<br />
βt − 1<br />
β − 1<br />
Henning Sulzbach J. W. Goethe-Universität Frankfurt a. M. <str<strong>on</strong>g>The</str<strong>on</strong>g> <str<strong>on</strong>g>C<strong>on</strong>tracti<strong>on</strong></str<strong>on</strong>g> <str<strong>on</strong>g>Method</str<strong>on</strong>g> <strong>on</strong> C([0, 1]) <strong>and</strong> D<strong>on</strong>sker’s <str<strong>on</strong>g>The</str<strong>on</strong>g>orem<br />
<br />
.