Strongly consistent autoregressive predictors in abstract Banach spaces

Abstract

This work derives new results on strong consistent estimation and prediction for autoregressive processes of order 1 in a separable Banach space B. The consistency results are obtained for the component-wise estimator of the autocorrelation operator in the norm of the space L(B) of bounded linear operators on B. The strong consistency of the associated plug-in predictor then follows in the B-norm. A Gelfand triple is defined through the Hilbert space constructed in Kuelbs' lemma [25]. A Hilbert-Schmidt embedding introduces the Reproducing Kernel Hilbert space (RKHS), generated by the autocovariance operator, into the Hilbert space conforming the Rigged Hilbert space structure. This paper extends the work of Bosq [10] and Labbas and Mourid [26].

Lagged Covariance and Cross-Covariance Operators of Processes in Cartesian Products of Abstract Hilbert Spaces

“…Probabilistic features of and estimators for lag-h-covariance operators C X;h of stationary processes X = (X k ) k∈Z with values in L 2 [0, 1], the space of measurable, square-Lebesgue integrable real valued functions with domain [0, 1], are widely studied for fixed lag h, see, e. g., [5], [19], [22], [34], [27]. Further, [39] developed covariance estimators in the space of continuous functions C[0, 1], [48] in tensor product Sobolev-Hilbert spaces, [33] for continuous surfaces, and [18], [1] for arbitrary separable Hilbert spaces. [34], [39], [18], [1] constrained their assertions to autoregressive (AR) processes, where [1] deduced the results for a random AR(1) operator.…”

“…Further, [39] developed covariance estimators in the space of continuous functions C[0, 1], [48] in tensor product Sobolev-Hilbert spaces, [33] for continuous surfaces, and [18], [1] for arbitrary separable Hilbert spaces. [34], [39], [18], [1] constrained their assertions to autoregressive (AR) processes, where [1] deduced the results for a random AR(1) operator. Thereby, [5], [19], [22], [1] utilized classical moment estimators, [27] estimated the integral kernels, in [18], [34] truncated spectral decompositions occured having estimated principle components, and [48] used operator regularized covariance estimators.…”

Functional ARCH and GARCH Models: A Yule-Walker Approach

“…Concerning parameter estimation in functional ARCH and GARCH processes, open problems are the estimation in general, separable Banach spaces, see Ruiz-Medina M. D. &Álvarez-Liébana J. [24], the asymptotic distribution of the estimations errors when estimating the parameters without projecting them on a finite-dimensional subspace, see [2] and [6] for the parameters projected on a finite-dimensinal subspace, and the asymptotic lower bounds of the estimations errors.…”

Functional ARCH and GARCH Models: A Yule-Walker Approach

“…can be defined (see [5]) and operator estimation was already executed (see e.g. [30]). Hereinafter, we putḢ := L 4 [0, 1], and for assertions regarding (G)ARCH we throughout impose the following.…”

“…We leave the investigations concerning probabilistic properties of (G)ARCH in general separable Banach spaces behind for future research, as we do for order estimation, see [21]. Concerning the parameters, unsolved problems are their estimation in Banach spaces (see [30]), the asymptotic distribution of their estimations errors (see [2], [6] for the parameters projected on a finite-dimensional subspace), and the asymptotic lower bounds of their estimations errors.…”