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

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

(5.70)

for almost all λ.

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

(5.71)

for almost all λ, provided that ν ≠ 0, where the a.e. continuity of W(f) is accounted for.

For n ≠ m, the left-hand side of (5.70) can be written as

(5.72)

for ν ≠ 0 and almost all λ, where we used the fact that, for n ≠ m, Ψ⁽ⁿ⁾(λ) = Ψ^(m)(λ) at most in a set of zero Lebesgue measure, and W(f) is a.e. continuous and regular as |f|→ ∞.

It can be easily verified that (5.70) holds for almost all λ also for ν = 0.

For those values of λ belonging to the set of zero Lebesgue measure such that Ψ⁽ⁿ⁾(λ) = Ψ^(m)(λ) for m I₀ (with I₀ containing n and depending on λ), one obtains

(5.73)

These values, however, give no contribution if the function in the lhs of (5.70) is integrated w.r.t. λ.

b. Let Ψ(λ) be a.e. derivable. It results that

(5.74)

for almost all λ₁ and λ₂.

Proof: It is similar to that of item (a) (see also Lemma 5.2.2).

c. Let Ψ_a(λ), Ψ_b(λ), and Ψ_c(λ) be a.e. derivable. It results that

(5.75)

for almost all λ₁ and λ₂.

Proof: By considering the Taylor series expansion for the functions Ψ_b, Ψ_a, and Ψ_c, we have a.e.

(5.76)

(5.77)

(5.78)

respectively. Thus,

(5.79)

from which (5.75) easily follows since W(∞) = 0.

d. Let Ψ_a(λ), Ψ_b(λ), and Ψ_d(λ) be a.e. derivable. It results

(5.80)

for almost all λ₁ and λ₂.

Proof: By considering the Taylor series expansion for the functions Ψ_b, Ψ_a, and Ψ_d, we have a.e.

(5.81)

(5.82)

(5.83)

respectively. Thus,

(5.84)

from which (5.80) easily follows since W(∞) = 0.

e. Let Ψ₄(λ₁, λ₂, λ₃) have a.e. all the partial derivatives. It results

(5.85)

for almost all λ₁ and λ₂, where ∇ is the gradient operator and superscript T denotes transpose.

Proof: By considering the Taylor series expansion of the function Ψ₄, one has a.e.

(5.86)

Thus,

(5.87)

from which (5.85) easily follows since W(∞) = 0.

5.5.1 Proof of Theorem 4.7.5 Asymptotic Expected Value of the Frequency-Smoothed Cross-Periodogram

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,

(5.89)

where, in the derivation of the second equality, Lemma 5.5.1a (with W = W_B) 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

(5.91)

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 δ_n−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.

5.5.2 Proof of Theorem 4.7.6 Rate of Convergence of the Bias of the Frequency-Smoothed Cross-Periodogram

For the sake of notation simplicity, let us put

From (4.150) with (4.151) substituted into we have

(5.93)

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.

• It results that a.e.

(5.96)

where with f_a min {f, f − ν/T}, f_b max {f, f − ν/T}, the assumption of bounded second-order derivative is used (Assumptions 4.4.4 and 4.5.4), and O_ν(·) denotes “big oh” Landau symbol depending on ν, that is, g(ν, T) = O_ν(1/T^a) means g(ν, T) ≤ K_ν/T^a as T→ ∞ with K_ν depending on ν. Since the second-order derivative is assumed to be uniformly bounded, then there exists K such that K_ν ≤ K, ∀ν. Thus, when necessary, O_ν(1) will be upper bounded by O(1).

• Using (5.96) one has a.e.

(5.97)

where with f_a min {f, f − λΔf − ν/T}, f_b max {f, f − λΔf − ν/T} and O_ν,λ(·) denotes “big oh” Landau symbol depending on ν and λ.

• Using (5.97) and considering the Taylor series expansion for , one obtains a.e.

(5.98)

provided that T→ ∞ and Δf → 0 with T(Δf)² → 0. In the second equality is appropriately chosen, and in the third equality the uniform boundedness of (Assumption 4.7.3) is exploited.

• For m ≠ n ( or ), accounting for (5.96) one obtains a.e.

• For , according to (4.156), it results that m ≠ n and Ψ^(m)(f) = Ψ⁽ⁿ⁾(f). Using (5.99) we have a.e.

(5.100)

provided that T→ ∞ and Δf → 0 with TΔf→ ∞ and T(Δf)² bounded, and γ is defined in Assumption 4.7.2. In (5.100), it is assumed that

(5.101)

That is, in the intersection points, curves have different slopes. Such a condition can be relaxed by assuming that there exists a derivative order p such that in the intersection points curves have equal derivatives up to order p − 1 and different pth-order derivatives.

The term in (5.100) cannot be easily managed in the bias expression. For this reason, the rate of convergence of bias is derived for all values of f such that the set is empty. That is, in all points f where two or more different curves do not intercept.

• For , m ≠ n, according to (4.156), it results that Ψ^(m)(f) ≠ Ψ⁽ⁿ⁾(f). Therefore, using (5.99) we have a.e.

• It results that a.e.

(5.103)

with appropriate , provided that the first-order derivative is bounded (Assumption 4.7.4).

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.

5.5.3 Proof of Theorem 4.7.7 Asymptotic Covariance of the Frequency-Smoothed Cross-Periodogram

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 ν_y1 and ν_x1 (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

(5.112)

Thus,

(5.113)

where, in the derivation of (5.113), Lemma 5.5.6c (with W = W_B, , , , and ) is used. Hence,

(5.114)

where Lemmas 5.5.1b (with W = W_A) 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

(5.115)

(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,

(5.116)

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 = W_B, , , , 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 ν_y1, ν_x1, and ν_y2 (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)