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Large and moderate deviations principles for functional autoregressive processes

We consider large and moderate deviations for the empirical mean and covariance of hilbertian autoregressive processes. As an application we obtain moderate deviation principles for the eigenvalues and associated projectors of the empirical covariance sequence. Keywords : Deviation principles, Hilbertian process, covariance operators, functional principal component analysis. 1 Introduction The study of random curves is a recent area in statistics. Its basic principles are the following. Statistical data are often discrete observations whereas the underlying phenomenon is usually continuous. Rather than dealing with the original data, an interesting approach consists in rebuilding a curve tting them, which will become the statistical observation. We refer to the monograph by Ramsay and Silverman [17] for a large review of functional techniques and numerous examples. The two main steps are, on a one hand, the reconstruction of the underlying processes (see the rst chapter of [17], and [20]) and, on the other hand, the statistical inference on these random curves. In the following, we will restrict ourselves to the second aspect. Suppose you are given a continuous time process = ( t)t2R. It is possible to associate with an innite sequence of random variables, namely Xn (t) = nT+t; where 0 t T; n 2 Z: Therefore (Xn) is a functional (i.e innite dimensional) discretetime process. It is possible to make statistical inference on Xn and hence on : In all the following, we will always deal with Hilbert spaces of functions (for instance, the space of all square integrable functions over [0; T ]). Focusing on statistical prediction of time-continuous processes, Bosq introduced the autoregressive hilbertian model (ARH(1)), which can be viewed as a generalization of the classical AR(1) model to the functional setting (see [6] for a recent review in this area). This model was successfully implemented in climatology (see e.g. [4] and [5]), medicine [3], and electricity consumption [7].