Mathematical Statistics and Stochastic Processes

Chapter 12

Square-Integrable Continuous-Time Processes

12.1. Definitions

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.

12.2. Mean-square continuity

(Xt) is said to be continuous in mean square at t0 ∈ T if t → t0 (t ∈ T) leads to E(X_t − X_t0)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.

PROOF.–

– (1) ⇒ (3) as (s, s′) → (t, t′) leads to and by the bicontinuity of the scalar product E(XsXs′) → E(XtXt′).

– (3) ...