3.6 Proofs for Section 2.4.3 “Asymptotic Normality of the Cyclic Cross-Correlogram”
In this section, proofs of results presented in Section 2.4.3 on the asymptotic Normality of the cyclic cross-correlogram are reported.
3.6.1 Proof of Lemma 2.4.17
By using (2.118), (2.125), and the multilinearity property of cumulants we have
where [−]i is an optional minus sign which is linked to the optional complex conjugation [*]i and in the third equality the variable changes uk = u, ui = u + si, i = 1, ..., k − 1 are made.
Thus,
where in the second inequality the variable change s = (u − tk)/T is made and Assumption 2.4.15 is used.
Therefore, from (3.126), accounting for Assumptions 2.4.5 and 2.4.15, it immediately follows that, for every k 
2 and every 
> 0, (2.168) holds.
The interchange of cum{ · } and integral operators in the second equality in (3.125) is allowed by the Fubini and Tonelli theorem (Champeney 1990, Chapter 3). In fact, by using Assumption 2.4.5 and the expression of a cumulant in terms of moments (1.209), (2.82b), the integrand function in the third term of equality (3.125) can be written as
Furthermore,
(3.127) 
where |μi| is the number of elements of μi, the fact that μi∩ μj = for i ≠ j is used and, accounting for Assumption 2.4.16, 
if μi 
{ 
1, ..., n}.
3.6.2 Proof of Theorem 2.4.18 Asymptotic Joint Normality of the Cyclic Cross-Correlograms
From Theorem 2.4.12 holding for 
we have
(3.128) 
From Theorem 2.4.13 it follows that the asymptotic covariance
(3.129) 
is finite. Analogously, from Theorem 3.7.2 it follows that the asymptotic conjugate covariance is finite. Moreover, from Lemma 2.4.17 with 
and k 3 in (2.168), we have
(3.130) 
Since the value of the cumulant does not change by adding a constant to each of the random variables (Brillinger 1981, Theorem 2.3.1), we also have
(3.131) 
That is, according to the results of Section 1.4.2, for every fixed αi, τi, ti, the random variables
i = 1, ..., k are asymptotically (T→ ∞) zero-mean jointly complex Normal (Picinbono 1996; van den Bos 1995).