ON LIMIT THEOREMS FOR BANACH-SPACE-VALUED LINEAR PROCESSES

Let (epsilon(i))(i is an element of Z) be i.i.d. random elements in a separable Banach space E, and let (a(i))(i is an element of Z) be continuous linear operators from E to a Banach space F such that Sigma(i is an element of Z) parallel to a(i)parallel to is finite. We prove that the linear process (X-n)(n is an element of Z) defined by X-n := Sigma(i is an element of Z) a(i)(epsilon(n-i)) inherits from (epsilon(i))(i is an element of Z) the central limit theorem and functional central limit theorems in various Banach spaces of F-valued functions, including Holder spaces.