In this section, proofs of lemmas and theorems presented in Section 4.7.2 on the asymptotic complex Normality of the frequency-smoothed cross-periodogram are reported.

**Fact 5.6.1**

*Let*

(5.122)

*where* [*]_{i} *represents the ith optional complex conjugation, and let us consider the k × 2 table*

*and a partition of its elements into disjoint sets* {ν_{1}, …, ν_{p}}. *The cumulant cum*{Z_{1}(f_{11}, f_{21}), …, Z_{k}(f_{1k}, f_{2k})} *can be expressed as (Leonov and Shiryaev 1959), (Brillinger 1965), (Brillinger and Rosenblatt 1967)*

*where* ν_{m} (m = 1, …, p) *are subsets of elements of the* k × 2 *table (5.123), is the cumulant of the elements in* ν_{m}, *and the (finite) sum in (5.124) is extended over all indecomposable partitions of table (5.123), including the partition with only one element (see the discussion following (2.159) for details).*

*Let*

(5.125)

*The elements of the table (5.123) can be identified by the pair ...*

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