5.4 Proofs for Section 4.6 “The Frequency-Smoothed Cross-Periodogram”

In this section, proofs of lemmas and theorems presented in Section 4.6 on bias and covariance of the frequency-smoothed cross-periodogram are reported.

5.4.1 Proof of Theorem 4.6.3 Expected Value of the Frequency-Smoothed Cross-Periodogram

By taking the expected value of the frequency-smoothed cross-periodogram (4.147) we have

(5.60)

from which (4.150) immediately follows.

In the third equality (4.106) is used. In the second equality, the interchange of expectation and convolution operations is justified by the Fubini and Tonelli Theorem (Champeney 1990, Chapter 3). In fact, defined

(5.61)

and accounting for Assumptions 4.4.3a, 4.4.5, and 4.6.2, for the integrand function in (5.60) we have

(5.62)

The interchange of sum and integral operations to obtain (4.150) from (5.60) is justified even if the set is not finite by using the dominated convergence theorem (Champeney 1990, Chapter 4). Specifically, by denoting with an increasing sequence of finite subsets of such that , we have

(5.63)

In fact, it results that

(5.64)

with the right-hand side bounded by the right-hand side of (5.62). That is, the integrand function in the second term of equality (5.63) is bounded by a summable function of (λ₁, ν₁) not depending on k.

5.4.2 Proof of Theorem 4.6.4 Covariance of the Frequency-Smoothed Cross-Periodogram

By setting

into (4.108) (with Δf = 1/T) we have (multilinearity property of cumulants (Mendel 1991))

(5.65)

where , , and are defined in (4.153), (4.154), and (4.155), respectively.

The interchange of cov{ · , · } and convolution operations and can be justified by the Fubini and Tonelli theorem. In fact, defined

(5.66)

and accounting for Assumptions 4.4.3a, 4.4.5, and 4.6.2, for the integrand function in (4.153) one obtains

(5.67)

An analogous result can be found for the integrand function in (4.154). Furthermore, defined the function

(5.68)

and accounting for Assumptions 4.4.3b, 4.4.5, and 4.6.2, for the integrand function in (4.155) one obtains

(5.69)

Finally, note that by using the dominated convergence theorem as in the proof of Theorem 4.6.3, it can be shown that in (4.153)–(4.155) the order of integral and sum operations can be interchanged.