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

Ion Grama ¹   ; Émile Le Page ¹   ; Marc Peigné ²

¹ Université de Bretagne Sud LMBA CNRS 6205 Campus de Tohannic BP 573 56017 Vannes Cedex, France
² Université F. Rabelais Tours LMPT CNRS 7350 Parc de Grandmont 37200 Tours Cedex, France

Colloquium Mathematicum, Tome 134 (2014) no. 1, pp. 1-55

Résumé

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.

Détail

DOI : 10.4064/cm134-1-1

Keywords: prove invariance principle non stationary random processes establish rate convergence under type mixing condition dependence exponentially decaying gap between past future controlled assumption characteristic function finite dimensional increments process distinctive feature mixing condition dependence increases exponentially dimension increments proposed mixing property particularly suited processes whose behavior described terms spectral properties related family operators several examples discussed work out explicit expressions constants involved bounds applied markov chains result specifies dependence constants properties underlying banach space initial state chain