An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

Abstract

Let $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ be the empirical process associated to an ℝ^d-valued stationary process (X _i ) _i≥0. In the present paper, we introduce very general conditions for weak convergence of $(U_{n}(t))_{t\in\mathbb{R}^{d}}$, which only involve properties of processes (f(X _i )) _i≥0 for a restricted class of functions $f\in\mathcal{G}$. Our results significantly improve those of Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011) and provide new applications.

The central interest in our approach is that it does not need the indicator functions which define the empirical process $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ to belong to the class $\mathcal{G}$. This is particularly useful when dealing with data arising from dynamical systems or functionals of Markov chains. In the proofs we make use of a new application of a chaining argument and generalize ideas first introduced in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011).

Finally we will show how our general conditions apply in the case of multiple mixing processes of polynomial decrease and causal functions of independent and identically distributed processes, which could not be treated by the preceding results in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011).