Strongly consistent autoregressive predictors in abstract Banach spaces

Model must align true,
Goodness-of-fit shows the way
Functional beauty
Banach spaces
Functional data
Functional time series

María Dolores Ruiz Medina, Javier Álvarez Liébana, «Strongly consistent autoregressive predictors in abstract Banach spaces», J. Multivariate Anal. 170, 186-201 (2019), doi: 10.1016/j.jmva.2018.08.001


Universidad de Granada

Universidad Complutense de Madrid


March 2019




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 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 (Kuelbs, 1970). 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 (2000) and Labbas and Mourid (2002).

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Cita BibTeX

  author = {M. D. Ruiz-Medina and J. Álvarez-Liébana},
  title = {Strongly consistent autoregressive predictors in abstract Banach spaces},
  journal = {J. Multivariate Anal.},
  volume = {170},
  pages = {186-201},
  keywords = {Banach spaces, continuous embeddings, functional plug-in predictors, strongly consistent estimators},
  doi = {10.1016/j.jmva.2018.08.001}
  url = {},
  year = {2019}