On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

Ion Grama, Émile Le Page, and Marc Peigné. "On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains." Colloquium Mathematicae 134.1 (2014): 1-55. <http://eudml.org/doc/284372>.

@article{IonGrama2014,
abstract = {We prove an invariance principle for non-stationary random processes and establish a rate of convergence under a new type of mixing condition. The dependence is exponentially decaying in the gap between the past and the future and is controlled by an assumption on the characteristic function of the finite-dimensional increments of the process. The distinctive feature of the new mixing condition is that the dependence increases exponentially in the dimension of the increments. The proposed mixing property is particularly suited to processes whose behavior can be described in terms of spectral properties of some related family of operators. Several examples are discussed. We also work out explicit expressions for the constants involved in the bounds. When applied to Markov chains, our result specifies the dependence of the constants on the properties of the underlying Banach space and on the initial state of the chain.},
author = {Ion Grama, Émile Le Page, Marc Peigné},
journal = {Colloquium Mathematicae},
keywords = {invariance principle; dependence measure; Markov chains; rate of convergence; mixing; spectral gap},
language = {eng},
number = {1},
pages = {1-55},
title = {On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains},
url = {http://eudml.org/doc/284372},
volume = {134},
year = {2014},
}

TY - JOUR
AU - Ion Grama
AU - Émile Le Page
AU - Marc Peigné
TI - On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains
JO - Colloquium Mathematicae
PY - 2014
VL - 134
IS - 1
SP - 1
EP - 55
AB - We prove an invariance principle for non-stationary random processes and establish a rate of convergence under a new type of mixing condition. The dependence is exponentially decaying in the gap between the past and the future and is controlled by an assumption on the characteristic function of the finite-dimensional increments of the process. The distinctive feature of the new mixing condition is that the dependence increases exponentially in the dimension of the increments. The proposed mixing property is particularly suited to processes whose behavior can be described in terms of spectral properties of some related family of operators. Several examples are discussed. We also work out explicit expressions for the constants involved in the bounds. When applied to Markov chains, our result specifies the dependence of the constants on the properties of the underlying Banach space and on the initial state of the chain.
LA - eng
KW - invariance principle; dependence measure; Markov chains; rate of convergence; mixing; spectral gap
UR - http://eudml.org/doc/284372
ER -