# 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 ϕ_{1} and ϕ_{2} are such that |ϕ_{1}| ≤ |ϕ_{2}|, ϕ_{1} is measurable and ϕ_{2} is integrable (i.e., ϕ_{2} is measurable and |ϕ_{2}| is integrable), then ϕ_{1} is integrable (Prohorov and Rozanov 1989, p. 82). Furthermore, if and , then |ϕ_{1}ϕ_{2}| ≤ |ϕ_{1}|||ϕ_{2}||_{∞} 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)

(3.38b)

(3.38c)

(3.38d)

from which ...