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# 5.2 Proofs for Section 4.4 “The Bifrequency Cross-Periodogram”

In this section, proofs of lemmas and theorems presented in Section 4.4 on bias and covariance of the bifrequency cross-periodogram are reported.

The following Lemma 5.2.1 allows, in the proofs of the subsequent lemmas and theorems, the free interchange of the order of limit and sum operations.

Lemma 5.2.1 (Napolitano 2003, Lemma A.1).

a. Under Assumption 4.4.3a (series regularity), the function series is uniformly convergent. Moreover, for any function sequence of uniformly bounded functions the function series is uniformly convergent.
b. Under Assumption 4.4.3b (series regularity), the function series

is uniformly convergent. Moreover, for any function sequence of uniformly bounded functions, the function series

is ...

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