Functional autoregressive process with seasonality

Abstract

We study the estimation of a seasonality perturbed by a continuous time process admitting a $C_{[0,\delta ]}$-valued autoregressive representation where $C_{[0,\delta ]}$ is the Banach space of continuous functions on $[0,\delta ],\delta >0.$ We provide the almost sure convergence, asymptotic normality and compact iterated logarithm law. Following Antoniadis A. (1982) we construct in the framework of functional autoregressive processes (non i.i.d. case), confidence balls for the seasonality in the space $C_{[0,\delta ]}$ from compact iterated logarithm law. Then when seasonality belongs to a finite dimensional space (dimensional reduction), we study the seasonality estimation giving its asymptotic properties. Finally, we examine an estimator of the dimension of this space when it is unknown. Numerical simulations illustrate the asymptotic results of the estimators.

Keywords:

• Autoregressive representation
• autoregressive process
• Banach space of continuous functions
• confidence ball
• dimensional reduction
• seasonality

Acknowledgment

The authors would like to thank the referees for their valuable comments which helped to improve the manuscript. This work is supported by D.G.R.S.D.T.