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

Tom 134 / 2014

Ion Grama, Émile Le Page, Marc Peigné Colloquium Mathematicum 134 (2014), 1-55 MSC: Primary 60F17, 60J05, 60J10; Secondary 37C30. DOI: 10.4064/cm134-1-1

Streszczenie

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.

Autorzy

  • Ion GramaUniversité de Bretagne Sud
    LMBA CNRS 6205
    Campus de Tohannic
    BP 573
    56017 Vannes Cedex, France
    e-mail
  • Émile Le PageUniversité de Bretagne Sud LMBA CNRS 6205
    Campus de Tohannic
    BP 573
    56017 Vannes Cedex, France
    e-mail
  • Marc PeignéUniversité F. Rabelais Tours
    LMPT CNRS 7350
    Parc de Grandmont
    37200 Tours Cedex, France
    e-mail

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