Hölderian invariance principle for Hilbertian linear processes

Račkauskas, Alfredas, and Suquet, Charles. "Hölderian invariance principle for Hilbertian linear processes." ESAIM: Probability and Statistics 13 (2009): 261-275. <http://eudml.org/doc/250626>.

@article{Račkauskas2009,
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>. },
author = {Račkauskas, Alfredas, Suquet, Charles},
journal = {ESAIM: Probability and Statistics},
keywords = {Central limit theorem in Banach spaces; Hölder space; functional central limit theorem; linear process; partial sums process; central limit theorem in Banach spaces; functional central limit theorem},
language = {eng},
month = {7},
pages = {261-275},
publisher = {EDP Sciences},
title = {Hölderian invariance principle for Hilbertian linear processes},
url = {http://eudml.org/doc/250626},
volume = {13},
year = {2009},
}

TY - JOUR
AU - Račkauskas, Alfredas
AU - Suquet, Charles
TI - Hölderian invariance principle for Hilbertian linear processes
JO - ESAIM: Probability and Statistics
DA - 2009/7//
PB - EDP Sciences
VL - 13
SP - 261
EP - 275
AB - 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>.
LA - eng
KW - Central limit theorem in Banach spaces; Hölder space; functional central limit theorem; linear process; partial sums process; central limit theorem in Banach spaces; functional central limit theorem
UR - http://eudml.org/doc/250626
ER -