In this section, proofs of lemmas and theorems presented in Section 4.7.1 on the mean-square consistency of the frequency-smoothed cross-periodogram are reported.

**Lemma 5.5.1** *Let* W(f) *be a.e. continuous and regular as* |f|→ ∞, *and* *(that is,* W *can be either* W_{A} *satisfying Assumption 4.4.5 or* W_{B} *satisfying Assumption 4.6.2). We have the following results*.

a. *Let* {Ψ^{(n)}(λ)} *be a set of a.e. derivable functions such that, for* n ≠ m, Ψ^{(n)}(λ) = Ψ^{(m)}(λ) *at most in a set of zero Lebesgue measure in . It results that*

for almost all λ, provided that ν ≠ 0, where the a.e. continuity of *W*(*f*) is accounted for.

for ν ≠ 0 and almost all

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