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Hölderian invariance principle for Hilbertian linear processes
Published online by Cambridge University Press: 04 July 2009
Abstract
Let $(\xi_n)_{n\ge 1}$ be the polygonal partial sums processes built
on the linear processes $X_n=\sum_{i\ge 0}a_i(\epsilon_{n-i})$
,
n ≥ 1, where $(\epsilon_i)_{i\in\mathbb{Z}}$
are
i.i.d., centered random elements in some
separable Hilbert space $\mathbb{H}$
and the ai's are bounded linear
operators $\mathbb{H}\to \mathbb{H}$
, with $\sum_{i\ge 0}\lVert a_i\rVert<\infty$
. We
investigate functional central limit theorem for $\xi_n$
in the
Hölder spaces $\mathrm{H}^o_\rho(\mathbb{H})$
of functions
$x:[0,1]\to\mathbb{H}$
such that ||x(t + h) - x(t)|| = o(p(h))
uniformly in t, where p(h) = hαL(1/h), 0 ≤ h ≤ 1
with 0 ≤ α ≤ 1/2 and L slowly varying at infinity. We
obtain the $\mathrm{H}^o_\rho(\mathbb{H})$
weak convergence of $\xi_n$
to
some $\mathbb{H}$
valued Brownian motion under the optimal assumption that
for any c>0, $tP(\lVert \epsilon_0\rVert>ct^{1/2}\rho(1/t))=o(1)$
when
t tends to infinity, subject to some mild restriction on L in
the boundary case α = 1/2. Our result holds in particular with
the weight functions p(h) = h1/2lnβ(1/h), β > 1/2>.
Keywords
- Type
- Research Article
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- Copyright
- © EDP Sciences, SMAI, 2009
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