## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

No credit card required

# 5.6 Proofs for Section 4.7.2 “Asymptotic Normality of the Frequency-Smoothed Cross-Periodogram”

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 {ν1, …, νp}. The cumulant cum{Z1(f11, f21), …, Zk(f1k, f2k)} can be expressed as (Leonov and Shiryaev 1959), (Brillinger 1965), (Brillinger and Rosenblatt 1967)

where νm (m = 1, …, p) are subsets of elements of the k × 2 table (5.123), is the cumulant of the elements in νm, 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 ...

## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

No credit card required