# 3.11 Proofs for Section 2.6.3 “Asymptotic Results as N→ ∞ and T_{s} → 0^{′′}

## 3.11.1 Proof of Lemma 2.6.12

Condition (2.209) holding uniformly w.r.t. τ assures that Assumption 2.6.11 is verified. The numbers M_{p}, possibly depending on , are independent of T_{s}. Under Assumption 2.6.11, the Weierstrass M-test (Johnsonbaugh and Pfaffenberger 2002, Theorem 62.6) assures the uniform convergence of the series of functions of T_{s}

Therefore, the limit operation can be interchanged with the infinite sum

(3.207)

where, in the second equality the sufficient condition (2.209) for Assumption 2.6.11 is used.

## 3.11.2 Proof of Theorem 2.6.13 Mean-Square Consistency of the Discrete-Time Cyclic Cross-Correlogram

From Lemma 2.6.12 we have

(3.208)

(not necessarily uniformly with respect to and m).

From Theorems 2.6.4 and 2.6.5 it follows that, for every fixed T_{s}, , and m,

(3.209)

that is,

(3.210)

Therefore, for ...