A functional limit theorem for η-weakly dependent processes and its applications

Abstract

Abstract. We prove a general functional central limit theorem for weak dependent time series. A very large variety of models, for instance, causal or non causal linear, ARCH(∞), bilinear, Volterra processes, satisfies this theorem. Moreover, it provides numerous application as well for bounding the distance between the empirical mean and the Gaussian measure than for obtaining central limit theorem for sample moments and cumulants.

Misparametrization subsets for penalized least squares model selection

“…We set the following assumptions on the X process : there exists q > 4 such that E[|X| q ] < ∞, and X is an η−weakly dependent process ( [8], [9]) such that the sequence η = (η r ) r∈N verifies: 0 < η r = O(r −α ) with α > max(3, 2m−1 m−4 ). This condition is not very restrictive and is satisfied, for example, for a Gaussian process whose covariance decreases exponentially.…”

Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate

“…As a particular case of the functional limit theorem presented in Bardet et al. (2008), a UCLT for a integrated periodogram can also be established, and more precisely a convergence rate to the Gaussian law.…”

“…As a particular case of the functional limit theorem presented in Bardet et al (2005) a UCLT for integrated periodogram can also be established, and more precisely a convergence rate to the Gaussian law Theorem 3 Let X = (X n ) n∈Z be η-weakly dependent process and such that assumption M holds. Suppose also…”

“…The second type of dependence under consideration is η ‐weak dependence. This property allows us to derive central limit theorems for non‐causal processes (see Bardet et al. , 2008 or Doukhan and Wintenberger, 2007).…”

“…The second type of dependence under consideration is η-weak dependence. This property allows to derive central limit theorems for non causal processes, see [2] or [19]. These results can be applied for instance, to two-sided linear or Volterra processes (see their definition below in Section 3).…”