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# 3.4 Proofs for Section 2.4.1 “The Cyclic Cross-Correlogram”

In this section, proofs of lemmas and theorems presented in Section 2.4.1 on the bias and covariance of the cyclic cross-correlogram are reported.

In the following, all the functions are assumed to be Lebesgue measurable. Consequently, without recalling the measurability assumption, we use the fact that if the functions ϕ1 and ϕ2 are such that |ϕ1| ≤ |ϕ2|, ϕ1 is measurable and ϕ2 is integrable (i.e., ϕ2 is measurable and |ϕ2| is integrable), then ϕ1 is integrable (Prohorov and Rozanov 1989, p. 82). Furthermore, if and , then |ϕ1ϕ2| ≤ |ϕ1|||ϕ2|| almost everywhere and, hence, .

## 3.4.1 Proof of Theorem 2.4.6 Expected Value of the Cyclic Cross-Correlogram

By using (2.31c) and (2.118) one has

(3.38a)

(3.38c)

from which ...

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