# Hölderian invariance principle for Hilbertian linear processes

Alfredas Račkauskas; Charles Suquet

ESAIM: Probability and Statistics (2009)

- Volume: 13, page 261-275
- ISSN: 1292-8100

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topRačkauskas, 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 -

## References

top- N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular variation. Encyclopaedia of Mathematics and its Applications. Cambridge University Press (1987). Zbl0617.26001
- J. Dedecker and F. Merlevède, The conditional central limit theorem in Hilbert spaces. Stoch. Process. Appl.108 (2003) 229–262. Zbl1075.60501
- J. Dedecker, P. Doukhan, G. Lang, J.R. Leon, S. Louhichi and C. Prieur, Weak Dependence: With Examples and Applications, volume 190 of Lect. Notes Statist. Springer (2007). Zbl1165.62001
- D. Hamadouche, Invariance principles in Hölder spaces. Portugal. Math.57 (2000) 127–151. Zbl0965.60011
- M. Juodis, A. Račkauskas and Ch. Suquet, Hölderian functional central limit theorems for linear processes. ALEA Lat. Am. J. Probab. Math. Stat.5 (2009) 47–64. Zbl1162.60317
- J. Kuelbs, The invariance principle for Banach space valued random variables. J. Multiv. Anal.3 (1973) 161–172. Zbl0258.60009
- J. Lamperti, On convergence of stochastic processes. Trans. Amer. Math. Soc.104 (1962) 430–435. Zbl0113.33502
- M. Ledoux and M. Talagrand, Probability in Banach Spaces. Springer-Verlag, Berlin, Heidelberg (1991). Zbl0748.60004
- F. Merlevède, M. Peligrad and S. Utev, Sharp conditions for the CLT of linear processes in a Hilbert space. J. Theoret. Probab.10 (1997) 681–693. Zbl0885.60015
- F. Merlevède, M. Peligrad and S. Utev, Recent advances in invariance principles for stationary sequences. Probab. Surveys3 (2006) 1–36. Zbl1189.60078
- A. Račkauskas and Ch. Suquet, Hölder versions of Banach spaces valued random fields. Georgian Math. J.8 (2001) 347–362. Zbl1017.60042
- A. Račkauskas and Ch. Suquet, Necessary and sufficient condition for the Hölderian functional central limit theorem. J. Theoret. Probab.17 (2004) 221–243. Zbl1069.60034
- A. Račkauskas and Ch. Suquet, Hölder norm test statistics for epidemic change. J. Statist. Plann. Inference126 (2004) 495–520. Zbl1084.62083
- A. Račkauskas and Ch. Suquet, Central limit theorems in Hölder topologies for Banach space valued random fields. Theor. Probab. Appl.49 (2004) 109–125. Zbl1095.60001
- A. Račkauskas and Ch. Suquet, Testing epidemic changes of infinite dimensional parameters. Stat. Inference Stoch. Process.9 (2006) 111–134. Zbl1110.62060
- M. Talagrand, Isoperimetry and integrability of the sum of independent Banach-space valued random variables. Ann. Probab.17 (1989) 1546–1570. Zbl0692.60016

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