| Abstract: | The autoregressive model in a Banach space (ARB) contains many continuous time processes used in practice, for example, processes that satisfy linear stochastic differential
equations of order k, a very particular case being the Ornstein–Uhlenbeck process. In this paper we study empirical estimators for ARB processes. In particular we show that, under some regularity conditions, the empirical mean is asymptotically optimal with
respect to a.s. convergence and convergence of order 2. Limit in distribution and the law of the iterated logarithm are also
presented. Concerning the empirical covariance operator we note that, if (X
n, n ∈ ℤ) is ARB then (X
n ⊗ X
n, n ∈ ℤ) is AR in a suitable space of linear operators. This fact allows us to interpret the empirical covariance operator as a sample mean
of an AR and to derive similar results for it.
This revised version was published online in August 2006 with corrections to the Cover Date. |
