Strong-consistent autoregressive predictors in abstract Banach spaces

01/26/2018

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by   M. D. Ruiz-Medina, et al.

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This work derives new results on the strong-consistency of a componentwise estimator of the autocorrelation operator, and its associated plug-in predictor, in the context of autoregressive processes of order one, in a real separable Banach space B (ARB(1) processes). For the estimator of the autocorrelation operator, strong-consistency is proved, 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 norm of B. The methodology applied is based on assuming suitable continuous embeddings between the Banach, Hilbert and Reproducing Kernel Hilbert spaces, involved in the construction proposed in Kuelbs Kuelbs70. This paper extends the results in Bosq00 and Labbas02.