5.5 Proofs for Section 4.7.1 “Mean-Square Consistency of the Frequency-Smoothed Cross-Periodogram”

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|→ ∞, img and img (that is, W can be either WA satisfying Assumption 4.4.5 or WB satisfying Assumption 4.6.2). We have the following results.

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

(5.70) equation

for almost all λ.
Proof: For n = m, the left-hand side of (5.70) can be written as

(5.71) equation

for almost all λ, provided that ν ≠ 0, where the a.e. continuity of W(f) is accounted for.
For nm, the left-hand side of (5.70) can be written as

(5.72) equation

for ν ≠ 0 and almost all

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