# 3.2 Proofs for Section 2.2.3 “Second-Order Spectral Characterization”

## 3.2.1 The μ Functional

In this section, the μ functional is introduced. It is defined as the functional whose value is the infinite-time average of the test function. The μ functional is formally characterized as the limit of approximating functions similar to the characterization of the Dirac delta as the limit of delta-approximating functions (Zemanian 1987, Section 1.3).

Let μ be the functional that associates to a test function ϕ its infinite-time average value. That is,

(3.16)

provided that the limit exists (and, hence, is independent of t).

In the following, the μ functional is heuristically characterized through formal manipulations. Let it be

(3.17)

where rect(t) = 1 for |t| ≤ 1/2 and rect(t) = 0 otherwise. For any finite T one has

(3.18)

Thus, in the limit as T→ ∞, we (rigorously) have

(3.19)

and we can formally write

(3.20)

with rhs independent of t, where μ(t) is formally defined as (see also (Silverman ...