Let (Xt, t ∈ T) be a real, square-integrable process defined on the probability space . We suppose T to be an interval of finite or infinite length on , and we set:
where m is the mean of (Xt) and C is its covariance.
In the following, unless otherwise indicated, we suppose that m = 0.
(Xt) is said to be continuous in mean square at t0 ∈ T if t → t0 (t ∈ T) leads to E(Xt − Xt0)2 → 0.
It is equivalent to say that is a continuous mapping from T to and that (Xt) is continuous in mean square (on all of T).
THEOREM 12.1.– The following properties are equivalent:
1) (Xt) is continuous in mean square.
2) C is continuous on the diagonal of T × T.
3) C is continuous on T × T.
– (1) ⇒ (3) as (s, s′) → (t, t′) leads to and by the bicontinuity of the scalar product E(XsXs′) → E(XtXt′).
– (3) ...