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Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle’s estimate. (English) Zbl 1198.62087

The authors prove uniform convergence results such as the strong law of large numbers and the central limit theorem for integrated periodograms of weakly dependent time series. The obtained results are applied to Whittle’s approximation likelihood estimate. Numerous articles have been written on this estimation method after Whittle’s seminal article. The asymptotic normality for Gaussian and causal linear, strong mixing and autoregressive conditionally heteroscedastic (ARCH)(\(\infty\)) processes was established by E.J. Hannan [J. Appl. Probab. 10, 130–145 (1973; Zbl 9261.62073); Corrections ibid., 913 (1973)], M. Rosenblatt [Stationary sequences and random fields. Boston etc.: Birkhäuser (1985; Zbl 0597.62095)], L. Giraitis and P.M. Robinson [Econom. Theory 17, No. 3, 608–631 (2001; Zbl 1051.62074)]. The main goal of the present article is to provide a unified treatment of the asymptotic normality for a very rich class of weakly dependent processes, including those previously mentioned, as well as for some new classes of non-causal or nonlinear processes.
Let \(X = (X_k)_{k\in\mathbb Z}\) be a real-valued zero-mean fourth-order stationary time series, \(R(s)\) the covariogram of \(X\), and let \(\kappa_4(i,j,k)\) be the fourth cumulants of \(X\). The authors use an assumption M on \(X\), which means that \(\gamma=\sum_{s\in\mathbb Z}(R(s))^2<\infty\) and \(\kappa_4=\sum_{i,j,k}| \kappa_4(i,j,k)| <\infty\). The periodogram of \(X\) is \(I_n(\lambda)=(2\pi n)^{-1}\left| \sum_{k=1}^nX_ke^{-ik\lambda}\right| ^2\) for a \(2\pi\)-periodic function such that \(g\in L^2(-\pi,\pi)\) defines the integrated periodogram of \(X\) and \(J_n(g)=\int_{-\pi}^{\pi}g(\lambda)I_n(\lambda)d\lambda\) and \(J(g)=\int_{-\pi}^{\pi}g(\lambda)f(\lambda)d\lambda\), where \(f\) denotes the spectral density of \(X\). The periodogram \(I_n(\lambda)\) could be a natural but not consistent estimator of the spectral density. At the same time the behaviour of \(J_n(g)\) becomes quite smooth, allowing estimation of the spectral density. A special case of the integrated periodogram is Whittle’s contrast, defined as a function \(\beta\to J_n(h_{\beta})\), where \(h_{\beta}\) is included in a class of functions depending on the vector of parameters \({\beta}\). The Whittle estimator minimizes this contrast. As a consequence, uniform limit theorems for the integrated periodogram \(J_n(\cdot)\) are the appropriate tools for obtaining uniform limit theorems for Whittle’s contrast, that imply limit theorems for Whittle’s estimators.
A uniform strong law of large numbers of integrated periodograms on a Sobolev-type space is first established only under the assumption M. Additional assumptions on the dependence properties of the time series have to be specified for establishing central limit theorems. The authors consider time series satisfying weak dependence properties introduced and developed by P. Doukhan and S. Louhichi [Stochastic Processes Appl. 84, No. 2, 313–342 (1999; Zbl 0996.60020)]. This frame of dependence includes a lot of models, like causal or non-causal linear, bilinear, strong mixing processes, and dynamic systems. The presented results are obtained under weaker conditions on time series, but considering different functional spaces compared to those obtained by R. Dahlhaus [ibid. 30, No. 1, 69–83 (1988; Zbl 0655.60033)] or T. Mikosch and R. Norvaiša [ibid. 70, No. 1, 85–114 (1997; Zbl 0913.60032)].
Two frames of weak dependence are considered. The first one exploits a causal property of dependence, the \(\theta\)-weak dependence property [see J. Dedecker and P. Doukhan, ibid. 106, No. 1, 63–80 (2003; Zbl 1075.60513)]. Under certain conditions, uniform limit theorems for the integrated periodogram and the asymptotic normality of Whittle’s estimate are established. These general results are new and extend Hannan’s and Rosenblatt’s classical results for causal linear or strong mixing processes. The second type of dependence under consideration is \(\eta\)-weak dependence. This property allows to derive central limit theorems for non-causal processes [see P. Doukhan and O. Wintenberger, Probab. Math. Stat. 27, No. 1, 45–73 (2007; Zbl 1124.60031)]. These results can be applied, for instance, to two-sided linear or Volterra processes.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M09 Non-Markovian processes: estimation
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
62M15 Inference from stochastic processes and spectral analysis

Keywords:

weak dependence
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