Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

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

(3.125)

where [−]_i is an optional minus sign which is linked to the optional complex conjugation [*]_i and in the third equality the variable changes u_k = u, u_i = u + s_i, i = 1, ..., k − 1 are made.

Thus,

(3.126)

where in the second inequality the variable change s = (u − t_k)/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 ...