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

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 nite dimensional increments of the process. The distinct 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 for 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 speci es the dependence of the constants on the properties of the underlying Banach space and on the initial state of the chain.
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https://hal.archives-ouvertes.fr/hal-00904518
Contributor : Ion Grama <>
Submitted on : Thursday, November 14, 2013 - 4:00:10 PM
Last modification on : Friday, June 28, 2019 - 10:04:02 AM

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  • HAL Id : hal-00904518, version 1

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Ion Grama, Emile Le Page, Marc Peigné. On the rate of convergence in the weak invariance principle for dependent random variables with application to Markov chains. Colloquium Mathematicum, 2014, 134 (1), pp.1-55. ⟨hal-00904518⟩

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