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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Lemma<br />
Properties of the Zolotarev distance ζs<br />
ζs is (s,+) - ideal, i.e.<br />
ζs(ϕ(X ), ϕ(Y )) ≤ ||ϕ|| s ζs(X , Y )<br />
for any c<strong>on</strong>tinuous <strong>and</strong> linear functi<strong>on</strong> ϕ : C([0, 1]) → C([0, 1]) with<br />
Furthermore<br />
||ϕ|| = sup<br />
f ∈C([0,1]),||f ||=1<br />
||ϕ(f )||.<br />
ζs(X1 + X2, Y1 + Y2) ≤ ζs(X1, Y1) + ζs(X2, Y2)<br />
for (X1, Y1) <strong>and</strong> (X2, Y2) independent.<br />
Proof: Zolotarev (’76)<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