In this paper, we study the problem of non parametric estimation of the stationary marginal density of an or a -mixing process, observed either in continuous time or in discrete time. We present an unified framework allowing to deal with many different cases. We consider a collection of finite dimensional linear regular spaces. We estimate using a projection estimator built on a data driven selected linear space among the collection. This data driven choice is performed via the minimization...
In this paper, we study the problem of non parametric estimation
of the stationary marginal density of an or a
-mixing process, observed either in continuous time or in
discrete time. We present an unified framework allowing to deal
with many different cases. We consider a collection of finite
dimensional linear regular spaces. We estimate using a
projection estimator built on a data driven selected linear space
among the collection. This data driven choice is performed the
minimization of...
Considering the centered empirical distribution function
as
a variable in , we derive non asymptotic upper
bounds for the deviation of the -norms of
as well as central limit theorems for the empirical process
indexed by the elements of generalized Sobolev balls. These results
are valid for a large class of dependent sequences, including
non-mixing processes and some dynamical systems.
In this paper we study the almost sure conditional central limit theorem in its functional form for a class of random variables satisfying a projective criterion. Applications to strongly mixing processes and nonirreducible Markov chains are given. The proofs are based on the normal approximation of double indexed martingale-like sequences, an approach which has interest in itself.
In this paper we derive the moderate deviation principle for stationary sequences of bounded random variables under martingale-type conditions. Applications to functions of -mixing sequences, contracting Markov chains, expanding maps of the interval, and symmetric random walks on the circle are given.
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