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

• Autorzy: Ion Grama , Émile Le Page , Marc Peigné
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems
• 60F17: Functional limit theorems; invariance principles
• 60J05: Discrete-time Markov processes on general state spaces
• 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2014
• Tom: 134
• Numer: 1
• Strony: 1-55

Daty

wydano

2014

Twórcy

autor

Ion Grama

• Université de Bretagne Sud, LMBA CNRS 6205, Campus de Tohannic, BP 573, 56017 Vannes Cedex, France

autor

Émile Le Page

• Université de Bretagne Sud LMBA CNRS 6205, Campus de Tohannic, BP 573, 56017 Vannes Cedex, France

autor

Marc Peigné

• Université F. Rabelais Tours, LMPT CNRS 7350, Parc de Grandmont, 37200 Tours Cedex, France

Identyfikatory

DOI

10.4064/cm134-1-1