3.1 Proofs for Section 2.2.2 “Second-Order Wide-Sense Statistical Characterization”
3.1.1 Proof of Theorem 2.2.17 Mean-Square Integrability of GACS Processes
By substituting t + τ = t1, t = t2 into (2.18) we have
(3.1)
Thus
(3.2)
where, in the first equality the variable changes t1 = t + τ and t2 = t are made, and in the last equality the result
(3.3)
is used. Thus, accounting for Assumption 2.2.16, the sufficient condition of Theorem 2.2.15 turns out to be satisfied.
3.1.2 Proof of Fact 2.2.18
where, in the second equality, the variable change t′ = t − τ is made. (3.4) is equivalent to (2.52) from which (2.54) follows.
From (2.54) we have
(3.5)
(3.6)
Therefore, (2.55) follows. In ...
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