Exponential Inequalities and Functional Estimations for Weak Dependent Data: Applications to Dynamical Systems

Abstract

We estimate density and regression functions for weak dependant datas. Using an exponential inequality obtained in [DeP] and in [Mau2], we control the deviation between the estimator and the function itself. These results are applied to a large class of dynamical systems and lead to estimations of invariant densities and on the mapping itself.2000 Mathematics Subject Classification. 37A50, 60E15, 37D20.

Adaptive density estimation under weak dependence

“…X t−1 = T (X t ) for some transformation T , namely T (x) = 2x1 1 0≤x<1/2 +(2x−1)1 1 1/2≤x≤1 . So called weak dependence coefficients have been recently developed to deal with such processes, see Dedecker et al [7], Maume-Deschamps [20], and references therein. Introduced by Dedecker and Prieur in [5], φ-weak dependence coefficients give sharp bounds on the covariance terms of dynamical systems, such as the stationary solution to (1.1).…”

“…Using these coefficients, we prove near-minimax results of thresholded wavelet estimators for dynamical systems called expanding maps. To our knowledge, only non adaptive density estimation has been studied in this non-mixing context, see for instance Bosq and Guegan [2], Prieur [22] and Maume-Deschamps [20]. The advantage of our approach is also to treat in one draw many other contexts of dependence.…”

Concentration of weakly dependent Banach-valued sums and applications to statistical learning methods

“…We use the standard notions of p-integrable and essentially bounded real functions spaces and use the notation L p (P) := L p (Ω, F , P) and L ∞ := L ∞ (Ω, F , P). Following [36], we define mixing processes with respect to a class a of real-valued functions. Let C(·) be a semi-norm over a closed subspace C of the Banach space of bounded real-valued functions f : X → R. We define the C-norm by f C := f + C(f ), where · is the supremum norm on C, and introduce…”

“…As discussed in [36], C-mixing describes many natural time-evolving systems and finds its application for a variety of dynamical systems. The authors of [26] use a slighty different definition of Φ C -mixing coefficient, where the supremum is taken over the class of functions {f : f C ≤ 1}.…”

“…This paper is organized as follows: in Section 2, we recall the setting for stochastic processes with values in a Banach space. We recall the definition from [36] to consider a general type of weakly dependent processes. In Section 3, we pose the main assumptions about the structure of the underlying infinite dimensional Banach space and present in a general form the new Bernstein-type inequalities for C−mixing processes.…”