The Contraction Method on C([0,1]) and Donsker's ... - Eurandom

<strong>The</strong>orem
Properties of ζs in C([0, 1])
Let B = C([0, 1]) and 0 < s ≤ 3, i.e, m ∈ {0, 1, 2}. Let (Xn)n≥1, X
be random variables in (C([0, 1]), · ) where, for each n ≥ 1, Xn is
piecewise linear with intervals of length at least rn. If
−m 1
ζs(Xn, X ) = o log , as n → ∞,
rn
then Xn → X in distribution.
Proof: This basically follows from a result of Barbour from the
context of Stein’s method.
Henning Sulzbach J. W. Goethe-Universität Frankfurt a. M. <strong>The</strong> <strong>Contraction</strong> <strong>Method</strong> on C([0, 1]) and Donsker’s <strong>The</strong>orem