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Mathematical Statistics and Stochastic Processes by Denis Bosq

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Chapter 12

Square-Integrable Continuous-Time Processes

12.1. Definitions

Let (Xt, tT) be a real, square-integrable process defined on the probability space images. We suppose T to be an interval of finite or infinite length on images, and we set:

images

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 t0T if tt0 (tT) leads to E(XtXt0)2 → 0.

It is equivalent to say that images is a continuous mapping from T to images 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 images and by the bicontinuity of the scalar product E(XsXs) → E(XtXt).

– (3) ...

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