3.12 Proofs for Section 2.6.4 “Concluding Remarks”
3.12.1 Proof of Theorem 2.6.21 Asymptotic Discrete-Time Cyclic Cross-Correlogram
Let t0 = n0Ts, T = (2N + 1)Ts fixed.
where
(3.242)
The stochastic function ψ(t) is mean-square Riemann-integrable in (t0 − T/2, t0 + T/2). That is, in the limit as the sampling period Ts approaches zero (and, hence, N→ ∞ so that (2N + 1)Ts = T is constant), in (3.241) we have
In fact, a necessary and sufficient condition such that (3.243) holds is (Loève 1963, Chapter X) (see also Theorem 2.2.15)
(3.244)
and the summability of can be proved under Assumptions 2.4.2–2.4.5 by following the proof of Theorem 2.4.7 (see (3.47b), (3.47c), and (3.52)–(3.54)).
Remark 3.12.1 Let
(3.245)
(3.246)
(3.247)
Then,
(3.248)
is not a useful bound to prove ...
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