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March 2014 On martingale approximations and the quenched weak invariance principle
Christophe Cuny, Florence Merlevède
Ann. Probab. 42(2): 760-793 (March 2014). DOI: 10.1214/13-AOP856


In this paper, we obtain sufficient conditions in terms of projective criteria under which the partial sums of a stationary process with values in $\mathcal{H}$ (a real and separable Hilbert space) admits an approximation, in $\mathbb{L}^{p}(\mathcal{H})$, $p>1$, by a martingale with stationary differences, and we then estimate the error of approximation in $\mathbb{L}^{p}(\mathcal{H})$. The results are exploited to further investigate the behavior of the partial sums. In particular we obtain new projective conditions concerning the Marcinkiewicz–Zygmund theorem, the moderate deviations principle and the rates in the central limit theorem in terms of Wasserstein distances. The conditions are well suited for a large variety of examples, including linear processes or various kinds of weak dependent or mixing processes. In addition, our approach suits well to investigate the quenched central limit theorem and its invariance principle via martingale approximation, and allows us to show that they hold under the so-called Maxwell–Woodroofe condition that is known to be optimal.


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Christophe Cuny. Florence Merlevède. "On martingale approximations and the quenched weak invariance principle." Ann. Probab. 42 (2) 760 - 793, March 2014.


Published: March 2014
First available in Project Euclid: 24 February 2014

zbMATH: 1354.60031
MathSciNet: MR3178473
Digital Object Identifier: 10.1214/13-AOP856

Primary: 60F15
Secondary: 60F05

Keywords: ergodic theorems , Martingale approximation , Moderate deviations , quenched invariance principle , stationary process , Wasserstein distances

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 2 • March 2014
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