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 WA satisfying Assumption 4.4.5 or WB satisfying Assumption 4.6.2). We have the following results.
(5.72)
(5.74)
(5.76)
(5.77)
(5.78)
(5.79)
(5.81)
(5.82)
(5.83)
(5.84)
(5.86)
(5.87)
By substituting (4.103) (with Δf = 1/T) and (4.148) into (4.150), we get
(5.88)
where, in the second equality, the variable change ν′ = (λ − ν)T is made. Moreover, here and in the following, for the sake of notation simplicity, we write
Thus,
where, in the derivation of the second equality, Lemma 5.5.1a (with W = WB) is used. Hence, (4.158) is proved.
In (5.89), in the second equality, the limit as T→ ∞ can be interchanged with the sum since the series of functions of T is uniformly convergent by the Weierstrass criterium (Smirnov 1964). In fact, let
(5.90)
one has
which is finite (and independent of T) due to Assumptions 4.4.3a, 4.4.5, and 4.6.2. In (5.91), the first L∞-norm is for functions of T and the others for functions of λ and ν′, respectively. Furthermore, the limit as T→ ∞ can be interchanged with the integral operation by the dominated convergence theorem (Champeney 1990, Chapter 4). In fact,
(5.92)
with the right-hand side independent of T. In (5.89), in the fourth equality (4.149) is accounted for and in the fifth equality the sampling property of Dirac delta is exploited. Furthermore, in the derivation of (5.89), the a.e. continuity of the functions and (Assumptions 4.4.3 a and 4.4.4) is used.
Finally, note that for those values of λ such that there exist m I0(λ) such that , the Kronecker delta δn−m in the second equality in (5.89) should be substituted by . However, since these values of λ belong to a set with zero Lebesgue measure (see the remark following Assumption 4.4.2) and the functions are not impulsive (Assumption 4.4.3a), then these values give zero contribution to the integral.
For the sake of notation simplicity, let us put
From (4.150) with (4.151) substituted into we have
where, in the third equality (4.103) and (4.107) (both with Δf = 1/T), (4.148) and (4.159) are used; in the fourth equality the order of integrals in dλ dν is interchanged and then the variable change ν′ = (λ − ν)T is made in the inner integral in dν (with λ fixed); in the fifth equality the variable change λ′ = (f − λ)/Δf is made in the integral in dλ and definition (4.156) of the set is accounted for.
In (5.93), the interchange of the order of integrals in dν′ and dλ′ can be justified by the Fubini and Tonelli theorem (Champeney 1990, Chapter 3). In fact, let
(5.94)
be the integrand function of the two-dimensional integral in (5.93). One has
(5.95)
where Assumptions 4.4.3 a (series regularity), 4.4.5 (data-tapering window regularity), and 4.6.2 (frequency-smoothing window regularity) are accounted for.
In the following, for notation simplicity, λ and ν will be used in place of λ′ and ν′.
Let us consider the following Taylor series expansions with Lagrange first-or second-order residual term. These expression, holding a.e., will be substituted into integrals. Consequently, the contribution of those points where expansions do not hold is zero.
By substituting (5.98) and (5.102)–(5.103) into (5.93), the bias for every f such that is empty can be expressed as sum of two terms and with the following asymptotic behaviors.
(5.104)
where, in the third equality the fact that and the identity
are used.
Furthermore, accounting for the bound
(5.105)
which is easily obtained since
(5.106)
and using the Taylor series expansions (5.97)–(5.103), the following upper bound is obtained.
(5.107)
provided that and , p = 1, 2, and (Assumptions 4.4.5, 4.7.2, 4.6.2, and 4.7.3).
(5.108)
One has
(5.109)
Thus,
(5.110)
from which (4.160) follows since γ ≥ 1.
Let us consider the term defined in (4.153). Accounting for (4.103), (4.107) (both with Δf = 1/T) and (4.148), it can be written as
(5.111)
Let us make the variable changes and in the inner integrals in νy1 and νx1 (λ1 and λ2 fixed) and then interchange the order of integrals so that the order is (from the innermost to the outermost) . Then, let us make the variable change in the inner integral in λ2 (λ1, , and fixed) and then interchange the order of the integrals in λ1 and λ2. Finally, let us make the variable change to obtain
Thus,
where, in the derivation of (5.113), Lemma 5.5.6c (with W = WB, , , , and ) is used. Hence,
where Lemmas 5.5.1b (with W = WA) and 5.3.5 are used. From (5.114), (4.162) immediately follows.
The interchange of the order of integrals can be justified by the Fubini and Tonelli theorem (Champeney 1990, Chapter 3). In fact, let be (TΔf) times the integrand function of the four-dimensional integral in (5.112). One has
(with the right-hand side independent of T and Δf), where Assumptions 4.4.3a, 4.4.5, and 4.6.2 have been accounted for. In (5.113), the interchange of the order of limit (as T→ ∞) and sum (over n′ and n′′) operations is justified since the series of functions of T
is uniformly convergent. In fact,
where the first L∞-norm is for functions of T. The right-hand side of (5.116) is bounded due to Assumptions 4.4.3 a, 4.4.5, and 4.6.2. Hence, the Weierstrass criterium (Smirnov 1964) can be applied. Moreover, from (5.115) it follows that (TΔf) times the integrand function in (5.112) is bounded by a summable function not depending on T. Thus, the dominated convergence theorem (Champeney 1990, Chapter 4) can be applied in (5.113) to interchange the order of limit and integral operations. As regards the derivation of (5.114), observe that is the integrand function of the four-dimensional integral in (5.113). In (5.114), the order of the limit (as Δf → 0) and sum (over n′ and n′′) operations can be interchanged since the series of functions of Δf
is uniformly convergent. In fact, for
where the L∞-norm is for functions of Δf, the same bound as in (5.116) can be obtained and, hence, the Weierstrass criterium can be applied. Moreover, the function is bounded by the function in the right-hand side of (5.115) which is summable and independent of Δf. Therefore, the dominated convergence theorem can be applied and the order of limit (as Δf → 0) and integral operations can be interchanged. Furthermore, in the derivation of (5.114), the a.e. continuity of the functions , , and , for z1, z2 {x, x*, y, y*}, (Assumptions 4.4.3 a and 4.4.4) are used.
Analogously, by considering the term defined in (4.154) and using Lemma 5.5.1d (with W = WB, , , , and ), it can be shown that
(5.117)
from which (4.163) immediately follows.
Let us now consider the term defined in (4.155). Accounting for (4.103), (4.107) (both with Δf = 1/T), and (4.148), it can be written as
(5.118)
Let us make the variable changes , , and in the inner integrals in νy1, νx1, and νy2 (λ1 and λ2 fixed). Then, let us make the variable changes and to obtain
(5.119)
Thus,
(5.120)
with the rhs bounded (Assumptions 4.4.5 and 4.7.3) and independent of T. Therefore,
(5.121)
3.144.28.70