2. ASSUMPTIONS

In this section, we state the assumptions used in the paper and relate them to the assumptions of Fox and Taqqu (1986) and Lieberman et al. (2003).

Throughout, s ≥ 3 denotes a positive integer associated with the order of an expansion. With conditions on derivatives up to order s + 1, we prove the validity of the (s − 2)th order formal Edgeworth expansion to the distribution of the WMLE with an error rate of o(n^−(s−2)/2).

Assumptions.

W2.

can be differentiated s + 1 times under the integral sign.

Assumption W1 is used because the WMLE is asymptotically normal only at points in the interior of Θ. Assumption W1 does not require Θ to be compact, as Fox and Taqqu (1986) do, because we do not prove consistency of the WMLE here.¹

Assumption W7 bounds the constants that appear in the preceding assumptions over parameter values θ that lie in compact sets. This assumption is needed to handle the remainder that appears in the approximation of the WMLE by a function of WLDs. It is also needed to deliver Edgeworth expansions that are valid uniformly over certain compact sets Θ* in Θ. Uniform results of this sort are required to establish the higher order improvements of the parametric bootstrap based on the WMLE.

Assumptions W1–W7 are satisfied for Gaussian ARFIMA(p,d,q) models.

3. PROPERTIES OF WLDs

In this section, we define the WLDs, specify the parameter values for which we can obtain Edgeworth expansions for WLDs and the WMLE, determine bounds on the magnitudes of the cumulants of the WLDs, and use these bounds to establish an Edgeworth expansion for the WLDs. This expansion is used in Section 4 to obtain an Edgeworth expansion for the WMLE.

3.1. Definition of WLDs

Let ν be a set of subscripts (r₁,…,r_q), where r_j is in {1,…,d_θ} for all j ≤ q. (We use r_j to denote an element of ν, rather than ν_j, because ν₁,…,ν_r are used subsequently to denote different vectors ν of subscripts.) Let D_ν L_n^W(θ) denote the qth order WLD with respect to θ specified by ν, namely,

In view of (3),

The second summand in (5) is

The last integral is the (j,k) element of the Whittle approximation to the inverse of the covariance matrix T_n(f_θ). More specifically, for an integrable function h on (−π,π), let T_n(h) denote the n × n Toeplitz matrix with (j,k) element

. Then, the Whittle approximation to T_n⁻¹(f_θ) is

For simplicity, we write

Then, the right-hand side (RHS) of (6) is

where M_n = I_n − P_n, P_n = n⁻¹1_n1_n′, and I_n is the identity matrix of order n. Because M_n1_n = 0,

Hence, without loss of generality, we may assume that μ = 0.

3.2. Parameter Values

We now specify the parameter values θ for which we establish Edgeworth expansions for WLDs and the WMLE that follows. Clearly, only parameter values that are in the interior of Θ and for which the asymptotic covariance matrix of the WMLE is nonsingular are candidates. For example, in an ARFIMA(p,d,q) model, parameter values θ for which there are common roots of the autoregressive and moving average characteristic equations are not candidates. Rather than excluding such parameter values from the parameter space Θ, which would not yield the parameter space typically used in practice, we allow the parameter space Θ to include such values, but we exclude them from the set of parameter values for which we establish an Edgeworth expansion.

By Theorem 2.1 of Dahlhaus (1989) (also see Fox and Taqqu, 1986, Theorem 2), the asymptotic covariance matrix of the WMLE (suitably normalized) is

By Dahlhaus (1989, Theorem 2.1 and Sect. 4), Σ(θ) is the inverse of the asymptotic information matrix.

Let Z_n(θ) denote 2n times the vector of all WLDs, D_ν L_n^W(θ), up to order s − 1. For example, for s = 2,

where ν_j = {j} for j = 1,…,d_θ and D_νj L_n^W(θ) = (∂/∂θ_j)L_n^W(θ). For s = 3, Z_n(θ) contains the partial derivatives given previously plus those corresponding to the following ν vectors: (1,1),(1,2),…,(1,d_θ), (2,2),(2,3),…,(2,d_θ),…,(d_θ,d_θ), where D_(i,j) L_n^W(θ) = (∂²/∂θ_i∂θ_j)L_n^W(θ).

Let D_n(θ) denote the covariance matrix of n^−1/2Z_n(θ) when the true parameter is θ. The (j,k) element of D_n(θ) is

(see (13), which follows). By Proposition 2 and Theorem 3, which follows, the asymptotic covariance matrix D(θ) (= lim_n→∞ D_n(θ)) of n^−1/2Z_n(θ) exists and its (j,k) element is

Given any subvector Z_n(θ) of Z_n(θ), let D_n(θ) and D(θ) denote the finite-sample and asymptotic covariance matrices of n^−1/2Z_n(θ), respectively, when the true parameter is θ.

We establish Edgeworth expansions for WLDs and the WMLE that hold uniformly over compact sets that lie in any set

that satisfies the following “nonsingularity” condition.

Condition NS.

Note that every parameter θ₁ that is in the interior of Θ and for which Σ(θ₁) is nonsingular is in some set

that satisfies condition NS. This follows because, given θ₁, there is a subvector of Z_n(θ), call it Z_θ1,n(θ), such that condition NS(iii) holds for θ ∈ {θ₁}. Hence, the set {θ₁} is an example of a set

that includes θ₁ and satisfies condition NS.

The first condition of condition NS(iii) is utilized because the vector of WLDs n^−1/2Z_n(θ) needs to have a nonsingular covariance matrix to apply Theorem 1 of Durbin (1980), which is used to obtain an Edgeworth expansion for the WLDs. For example, in an ARFIMA(1,d,1) model, the third derivative with respect to the autoregressive parameter is zero. Hence, Z_n(θ) does not contain this WLD for any set

that satisfies condition NS.

In some models, the subvector Z_n(θ) of Z_n(θ) that yields a nonsingular asymptotic covariance matrix D(θ) in condition NS(iii) depends on the parameter vector θ. For example, in an ARFIMA(1,d,1) model, the first partial derivatives with respect to the autoregressive and moving average coefficients are linearly independent for most parameter values. But, at parameter values that yield common roots, these two WLDs are equal. The results given subsequently cover such cases by allowing one to consider different sets

, which may have different subvectors Z_n(θ) appearing in condition NS(iii).

The second condition of condition NS(iii) guarantees that the finite-sample covariance matrix of any subvector

that strictly contains Z_n(θ) is singular (as shown in the next paragraph). This is important because we obtain a valid Edgeworth expansion to the WMLE by approximating it by a function of Z_n(θ) (suitably normalized). If there is a subvector

that strictly contains Z_n(θ) and has a nonsingular covariance matrix, then Z_n(θ) does not contain all of the nonredundant WLDs of order up to s − 1 for sample size n. Nonredundant WLDs cannot be omitted from the approximation to the WMLE without affecting the accuracy of the approximation and the remainder of the Edgeworth expansion for the WMLE.

To see why the claim in the first sentence of the previous paragraph is true, let

denote the finite-sample and asymptotic covariance matrices of

, respectively. Let ν₁,…,ν_ds denote the derivative indices corresponding to the elements of

. By condition NS(iii),

is singular. Hence, by (11), there exists a vector a = (a₁,…,a_ds)′ ≠ 0 such that

for all λ in a subset of (−π,π) with Lebesgue measure 2π (using the fact that f_θ(λ) > 0 for all λ ≠ 0 for all θ by Assumption W3).

From (5), the elements of

are of the form

for

. Hence, (12) implies that

. In turn, this implies that

is singular.

The situation is different when establishing an Edgeworth expansion for the MLE, rather than the WMLE, as in Lieberman et al. (2003). In this case, one approximates the MLE by a vector of LLDs. Singularity of the asymptotic covariance matrix of a vector of LLDs does not imply singularity of the corresponding finite-sample covariance matrix. For example, with the ARFIMA(1,d,1) model, one can have a singular asymptotic covariance matrix of LLDs but a nonsingular finite-sample covariance matrix; see Section 5 for a discussion of this model. In such cases, when one approximates the MLE by a vector of LLDs whose asymptotic covariance matrix is nonsingular, some LLDs that are not redundant in finite samples are omitted. This affects the accuracy of the approximation of the MLE and the remainder of the Edgeworth expansion of the MLE. In consequence, in models with this feature, the Edgeworth expansion for the MLE of Lieberman et al. (2003) has remainder o(n⁻¹), rather than o(n^−(s−2)/2).

Let

Subsequently we establish an Edgeworth expansion for the vector W_n(θ) of normalized WLDs. Denote the dimension of Z_n(θ) and W_n(θ) by d_s. The elements of W_n(θ) are of the form

Note that, although Assumptions W2–W7 concern derivatives up to order s + 1, for an Edgeworth expansion to the WMLE with an error rate of order o(n^−(s−2)/2), we only need an Edgeworth expansion of the joint distribution of a vector of normalized WLDs up to order s − 1, namely, W_n(θ). The reason is that, in the Taylor series approximation of the WMLE by WLDs, the (s + 1)th order WLDs are in the remainder term and the sth order WLDs can be replaced in the series expansion by their expectations with the differences between them and their expectations being added to the remainder term. For example, see Taniguchi and Kakizawa (2000, Sect. 4.2).

3.3. WLD Cumulant Bounds

A key step in establishing the validity of an Edgeworth expansion for the distribution of W_n(θ) that holds uniformly over compact subsets of some set

that satisfies condition NS is showing that the cumulants of Z_n(θ) are O(n) uniformly in such sets. This condition is Assumption 4 of Durbin (1980). Durbin's Theorem 1 is used to obtain the Edgeworth expansion of W_n(θ).

Let κ_r(θ) denote an rth order joint cumulant of Z_n(θ). For simplicity, we drop the subscript n in the following. From the theory of quadratic forms in normal variables (e.g., see Searle, 1971, p. 55), κ_r(θ) can be written as

for some vectors {ν_j : j = 1,…,r} of subscripts and some constant C_r < ∞. Note that κ_r(θ) involves derivatives of f_θ⁻¹(λ), not of f_θ(λ).

To clarify the notation, note that Z_n(θ) is a vector whose elements are partial derivatives of L_n^W(θ) of order s − 1 or less. For example, the jth element of Z_n(θ) might be D_νj L_n^W(θ) = (∂²/∂θ₁∂θ₂)L_n^W(θ), where ν_j = (1,2)′. Now, given the vector Z_n(θ) of partial derivatives, an rth order cumulant of Z_n(θ), like an rth order moment of Z_n(θ), is determined by r elements of Z_n(θ) with repeated elements allowed.

THEOREM 1. Suppose Assumptions W1–W7 hold and

satisfies condition NS. Then, for all r ≥ 1, κ_r(θ) = O(n) uniformly over any compact subset Θ* of

.

To prove Theorem 1, we substitute M = I − P in (13) and rewrite κ_r(θ) for r ≥ 2 as

where χ_j,ξ_j take on the values zero or one and satisfy

, the summation is over all 2^2r possible configurations of (χ₁,ξ₁,…,χ_r,ξ_r), and (−P)⁰ = I. The following result establishes that the summand in (14) for which χ_j = ξ_j = 0 for all j = 1,…,r is O(n). The result is due to Lieberman et al. (2003) (see their Theorem 1). It is a uniform version of Theorem 1.a of Fox and Taqqu (1987).

PROPOSITION 2. Suppose Assumptions W1–W7 hold and

satisfies condition NS. Then, for all r ≥ 1,

for any compact subset Θ* of

.

In Proposition 2, Assumption W2 guarantees the existence of the WLDs, Assumptions W3 and W4 specify the exponent structure of the matrix product, Assumptions W5 and W6 are used in the proof of Theorem 1 of Lieberman et al. (2003), and Assumption W7 is required for the result to be uniform. Proposition 2 is used to show that the rth order cumulants are O(n). It is not used to approximate the cumulants by their limiting values up to the order of the Edgeworth expansion given subsequently because the Edgeworth expansion is given in terms of the finite-sample cumulants.

Next, we consider the case where at least one matrix P appears in (14). Because P is of the form n⁻¹11′ (where 1 denotes an n-vector of ones), for any matrices A and B, tr[PAPB] = tr[PA] tr[PB]. In consequence, each summand in (14) for which at least one matrix P appears can be written as the product of terms of the following form for different values of p: for 0 ≤ p ≤ r,

In addition, the number of terms of the form I_n,p⁻(θ) and I_n,p⁺(θ) that appear in each summand must be the same, because each product in (14) must contain the same number r of matrices T(f_θ) as matrices of the form T(g_θ,νj).

For example, if r = 2, then a typical term in the sum in (14) that contains at least one P matrix is of the following form: for (χ₁,ξ₁,χ₂,ξ₂) = (1,1,1,1), with P = n⁻¹11′, T₁ = T(g_θ,ν1), T₂ = T(g_θ,ν2), and T = T(f_θ) for brevity,

which is a product of terms of the form I_n,0⁻(θ), I_n,0⁺(θ), I_n,0⁻(θ), and I_n,0⁺(θ). Or, if (χ₁,ξ₁,χ₂,ξ₂) = (1,0,1,0),

which is a product of two terms of the form I_n,1(θ).

The following theorem makes extensive use of power counting theory as discussed in Fox and Taqqu (1987). The theorem is analogous to Theorem 1(a) of Fox and Taqqu (1987). Its proof is complicated by the fact that the algebraic structure of the product matrices is not homogeneous.

THEOREM 3. Suppose Assumptions W1–W7 hold and

satisfies condition NS. For any p ≥ 0, any compact set

, and any constant δ > 0, there exists a constant K_p(Θ*,δ) < ∞ such that

(a) sup_θ∈Θ*|I_n,p(θ)| ≤ K_p(Θ*,δ)n^δ,

(b) sup_θ∈Θ*|I_n,p⁻(θ)| ≤ K_p(Θ*,δ)n^−α+δ, and

(c) sup_θ∈Θ*|I_n,p⁺(θ)| ≤ K_p(Θ*,δ)n^α+δ.

Proposition 2 shows that the summand in (14) in which no P matrix appears is O(n). Every other summand in (14) is a product of terms in (16), which by Theorem 3 is O(n^δ) (using the fact that the I_n,p⁻(θ) and I_n,p⁺(θ) terms come in pairs). Hence, the sum of terms in (14), namely, κ_r(θ), is O(n), and Theorem 1 holds.

3.4. WLD Edgeworth Expansion

We now state the Edgeworth expansion for the density of W_n(θ). It is obtained by applying Theorem 1 of Durbin (1980).

THEOREM 4. Suppose Assumptions W1–W7 hold and

satisfies condition NS. For

, let G_n(u,θ) be the joint density of W_n(θ) and let

be its (τ − 2)-order formal Edgeworth expansion for any integer τ ≥ 3. Then,

uniformly over u ∈ R^d_s and θ in any compact subset Θ* of

.

The Edgeworth expansion for the density of W_n(θ) can be used to obtain an Edgeworth expansion for the distribution of W_n(θ) using Corollary 3.3 of Skovgaard (1986).

COROLLARY 5. Suppose Assumptions W1–W7 hold and

satisfies condition NS. For any integer τ ≥ 3,

uniformly over all Borel sets C and θ in any compact subset Θ* of

.

Note that the preceding Edgeworth expansions are valid to any order under Assumptions W1–W7. (In fact, this feature of Theorem 4 is used to obtain the error o(n^{−(τ−2)/2}) in Corollary 5, rather than o(n^{−(τ−2)/2+δ}) for arbitrary δ > 0, when applying Skovgaard's result.) That is, the integer s determines the order of the WLDs that appear in W_n(θ) and, hence, Assumptions W1–W7 depend on s. But, given s and W_n(θ), the Edgeworth expansion in Theorem 4 is valid to any order τ ≥ 3.

4. EDGEWORTH EXPANSIONS FOR THE WHITTLE MLE

The WMLE

of θ solves

In general, there may be multiple solutions to (20).

Let

be the (s − 2)th-order formal Edgeworth expansion of the density of

given by

where φ_Σ(θ) denotes the multivariate normal density with mean zero and covariance matrix Σ(θ) and {q_n,r,θ(u) : r = 3,…,s} are Edgeworth polynomials whose coefficients are O(1) and depend on the cumulants of the WLDs.

The main result of the paper is the following theorem. Its form is analogous to that of Theorem 3 of Bhattacharya and Ghosh (1978).

THEOREM 6. Suppose Assumptions W1–W7 hold and

satisfies condition NS. Let Θ* denote a compact set in

. Then,

(a) there exists a sequence of estimators

and a constant d₀ = d₀(Θ*) such that

(b) any sequence of estimators

that satisfies (21) admits the Edgeworth expansion

uniformly over θ ∈ Θ* and over every class

of Borel sets that satisfies the condition

where (∂C)^ε denotes the ε-neighborhood of the boundary of C.

Several remarks are in order. First, the error rate in both parts of the theorem is identical to the i.i.d. rate. Second, the error rate can be made arbitrarily small if the assumptions hold for s arbitrarily large. Third, part (a) of the theorem does not guarantee consistency of the estimator that maximizes the Whittle likelihood. Rather, it shows that a consistent solution to the first-order conditions given in (20) exists. This is analogous to the results of Bhattacharya and Ghosh (1978). Fourth, the regularity condition in (23) is standard. It is the same as in Bhattacharya and Ghosh (1978, equation (1.6)).

5. AN EXAMPLE

The ARFIMA(1,d,1) model is very popular in applied work because of its flexibility. The model is

where B is the lag operator, d ∈ (0,½) is the long-memory parameter, and |φ| < 1 and |ψ| < 1 for stationarity and invertibility. Let θ = (d,φ,ψ,σ_ε²)′, where σ_ε² is the variance of the innovation ε_t.

The spectral density of the ARFIMA(1,d,1) process is

(e.g., see Hosking, 1981). The spectral density satisfies

where α(θ) = d, which is a special case of (1).

In this model, the third partial derivative of f_θ⁻¹(λ) with respect to (w.r.t.) φ is zero. In consequence, by the argument in Section 3.2, the matrices D(θ) and D_n(θ) are both singular. Because the same degeneracy occurs in D(θ) and D_n(θ), the problematic WLD can be deleted from Z_n(θ) without affecting the error in the approximation of the WMLE by the vector of WLDs. That is, for any set

that satisfies condition NS, the vector Z_n(θ) does not include the problematic WLD and this singularity does not cause a problem.

In contrast, when deriving an Edgeworth expansion for the MLE, one considers LLDs rather than WLDs; the asymptotic covariance matrix of an LLD vector that includes the third derivative w.r.t. φ is singular, but its finite-sample covariance matrix is nonsingular whenever the submatrix without the third derivative w.r.t. φ is nonsingular. This occurs because (i) the covariance matrix of all the LLDs up to order s − 1 is the same as D_n(θ) defined in (10), but with T_n(g_θ,νj) replaced by D_νjT_n⁻¹(f_θ), (ii) the limit as n → ∞ of the covariance matrix of all the LLDs up to order s − 1 is exactly the same as that of D_n(θ), namely, D(θ) (see Lieberman et al., 2003), and (iii) the third partial derivative of T_n⁻¹(f_θ) w.r.t. φ does not equal zero. In consequence, when one drops the third partial derivative w.r.t. φ from the vector Z_n(θ) that is used to approximate the MLE, the approximation of the MLE and the remainder of the Edgeworth expansion are affected.

6. PROOFS

Proof of Theorem 3. We prove the results of the theorem for the case where p ≥ 1 first. The proof closely follows the work of Fox and Taqqu (1987) using power counting theory. We use their notation. For ease of presentation, we omit the δ from the exponents in the bounds on f and the g_νj's. The proof goes through with δ added for some δ > 0 sufficiently small. Then, the results of the theorem hold for arbitrary δ > 0 because the upper bounds in the theorem are increasing in δ.

First, we consider I_n,p(θ). We have

where

Note that

where

In the case of Fox and Taqqu (1987), where the matrix P does not appear in the product, the function P_n(y) is given by

Hence,

differs from P_n(y) in that the term h_n*(−y_2p)h_n*(y₁) appears in place of h_n*(y₁ − y_2p). In addition, the RHS of (24) contains the n⁻¹ multiplicand, which is not present in the case of Fox and Taqqu (1987). For each h_n*(z), we use the following bound: for all 0 < η < 1,

where

(see Fox and Taqqu, 1987, p. 227). It follows that

for some constant K < ∞, for all θ ∈ Θ*, where

and α = α(θ).

We make the following change of variables:

Then, the RHS of (26) is at most

where

Notice that (−π ≤ y₁ ≤ π) ⇒ (−π ≤ x₁ ≤ π) and (−π ≤ y_2p ≤ π) ⇒ (−π ≤ x₁ + ··· + x_2p ≤ π). Hence, for all 0 < γ₁ < 1 and 0 < γ₂ < 1, we have

For all other h_n,η's appearing in (27), we have −2π ≤ x_k ≤ 2π for k = 2,…,2p, and so we need to consider the possibility that some of the ξ's, defined in (25), are not zero. Thus, the term in (27) is dominated by

where

The idea is to provide conditions on γ₁ and γ₂ such that the integral in (28) is finite.

It is useful to rewrite

as

where

and

To proceed, we distinguish between two cases.

Case I. ξ₂ = ξ₃ = ··· = ξ_2p = 0. Let

where

Section 5 of Fox and Taqqu (1987) shows that P_η(x) is integrable provided

The integrand

appearing in (28) and defined in (29) differs from P_η(x) in two respects.

Difference (D1).

.

Difference (D2). The exponent structure of P₂(x) is E₂ = {−α,−β,…,−α,−β}, whereas that of

is

Although the exponent structure of P₂(x) is homogeneous, the exponent structure of

is not homogeneous. This is important, because the conditions in (33) are not sufficient for integrability in the nonhomogeneous case. Moreover, it is clear from the proof of Proposition 5.5 of Fox and Taqqu (1987) that the extension of condition (33) to nonhomogeneous exponent structures is not trivial.

Our goal is to show that the function

is integrable by accommodating for the differences (D1) and (D2). The first difference leads to a simplification, whereas the latter leads to a complication. We deal first with (D1). It is clear from the discussion on page 222 of Fox and Taqqu (1987) that it is enough to consider sets W ⊂ T that do not contain x₂ + ··· + x_2p, where the set of functions T in Fox and Taqqu (1987) is given by

Note that T is the set of multiplicands of P_η(x) without the exponents or absolute values. The analogous set in our case is

For any W ⊂ T, let s(W) = T ∩ span(W). Although it is enough to consider the integrability of P_η(x) with the restriction x₂ + ··· + x_2p ∉ W, Fox and Taqqu (1987) still need to consider the case x₂ + ··· + x_2p ∈ s(W). For any

. Because

, it is also true that

. Hence, unlike the situation in Fox and Taqqu (1987), we do not need to consider the case

.

As in Section 3 of Fox and Taqqu (1987), we define the dimension of P_η,γ w.r.t. a set

to be

where |W| denotes the cardinality of W and the b_j's and L_j's are defined in (31) and (32). One can think of the elements of

arranged in columns as follows:

The top element in each column arises from the bound on P_n(x), and the bottom element in each column, apart from the first and last columns, arises from the bound on Q(x). As in Fox and Taqqu (1987, p. 223), we consider a set W that contains at most one element from each column. We partition W into contiguous “blocks” such that

, where a set B ⊂ W is a “block” if there exist [ell ]_B < r_B such that (i) W contains neither column [ell ]_B − 1 nor column r_B + 1 and (ii) B contains column [ell ]_B through r_B and no other columns. As in Fox and Taqqu (1987), the integral in (28) is finite if

for all B_i.

Let B denote one of the blocks B_i. It contains a block of columns l through r, so |B| = r − l + 1. Let m denote the smallest k satisfying x₁ + ··· + x_k ∈ B. We can write

, where

We count the powers associated with W₁ and obtain (r − l)(η − 1). Similarly, the powers associated with W₂ contribute

, where the α_j's are defined in (32). Thus,

Sufficient conditions for

to be positive are

Recall that

Hence, the second condition in (35) is satisfied if

because inf_θ∈Θ* α = inf_θ∈Θ* α(θ) > 0. Returning to (28), we see that the entire expression is at most K_p(Θ*,δ)n^δ for some constant K_p(Θ*,δ) < ∞ for all θ ∈ Θ*, ∀δ > 0, because sup_θ∈Θ* α = sup_θ∈Θ* α(θ) < 1.

As noted previously, we do not need to consider the case

, so the proof is complete for the ξ₂ = ··· = ξ_2p = 0 case.

Case II. At least one ξ_j ≠ 0 for j = 2,…,2p. This case is dealt with in Section 6 of Fox and Taqqu (1987). It is clear from (30) and (31) that the only L's affected in this case are L₂,…,L_2p. In the case of Fox and Taqqu (1987), the L's affected are L₁,…,L_2p, each of which has exponent η − 1. In their case, Fox and Taqqu (1987) fix a permutation {σ₁,…,σ_4p} of {1,…,4p} and define

A basis is constructed for T satisfying

By the proofs of Propositions 6.1 and 6.2 of Fox and Taqqu (1987), it follows that

provided the conditions in (36) hold. Because U_π = ∪_σ E_σ^π, we are done in this case also.

Next, we consider I_n,p⁻(θ). We have

where in this case U_π = [−π,π]^2p+1,

The exponent structure in this case is

Note that here we choose γ₁ = γ₂ = γ in the first and last terms in

. The second condition in (35) is satisfied if we take 2γ > 1 − α. Together with the condition that 0 < γ < 1, it follows that sup_θ∈Θ*|I_n,p⁻(θ)| ≤ K_p(Θ*,δ)n^−α+δ for some constant K_p(Θ*,δ) < ∞, ∀δ > 0.

Finally, the proof for the bound on I_n,p⁺(θ) uses the same ideas as previously. In this case, the exponent structure is

Sufficient conditions for integrability are 0 < γ < 1 and 2γ > 1 + α, from which it follows that sup_θ∈Θ*|I_n,p⁺(θ)| ≤ K_p(Θ*,δ)n^α+δ, ∀δ > 0. The proof of Theorem 3 is now complete for the case where p ≥ 1.

To finish the proof, we consider the case where p = 0. We have I_n,0(θ) = tr[P] = 1, so part (a) of the theorem holds trivially. Next, we have

where 1 denotes an n-vector of ones. We use the inequality

(see Fox and Taqqu, 1987, pp. 227, 237). Because |g_θ,ν1(λ)| ≤ c₂(θ,δ)|λ|^α, ∀λ ∈ N_δ, ∀δ > 0, where α = α(θ) − δ, by Assumption W4, and

by Assumption W2, there exists a constant K < ∞ such that

The integral is finite provided 2(η − 1) + α > −1 or 2η − 1 > −α. Take η such that 2η − 1 is arbitrarily close to −α to obtain the desired result.

Similarly, I_n,0⁺(θ) = tr[PT(f_θ)] = n⁻¹1′T(f_θ)1. By Assumption W3, | f_θ(λ)| ≤ c₂(θ,δ)|λ|^α, ∀λ ∈ N_δ, ∀δ > 0, where α = −α(θ) − δ. The preceding argument for I_n,0⁻(θ) holds for both positive and negative α, as long as |α| < 1. Hence, the desired result holds for I_n,0⁺(θ). █

Proof of Theorem 4. Theorem 4 is proved by verifying Assumptions 1–4 of Theorem 1 of Durbin (1980). Durbin's Assumption 1 corresponds to our Condition NS. Durbin's Assumption 4 requires that the joint cumulants of the WLDs are O(n) uniformly on Θ*. This is satisfied by our Theorem 1. Durbin's Assumptions 2 and 3 concern the characteristic function of Z_n(θ), which we denote by φ_n(ω,θ) = E_θ [exp(iω′Z_n(θ))]. From standard theory on quadratic forms in Gaussian variables (see, e.g., Searle, 1971, p. 55), we have

where A_j(θ) is a partial derivative of g(θ) (defined in Assumption W2) of order less than or equal to s − 1, B_j(θ) = MT(g_θ,νj)M, g_θ,νj is a partial derivative of (4π²)⁻¹f_θ⁻¹(λ) of order less than or equal to s − 1, and

. Because Proposition 2 and Theorem 3 ensure that

uniformly on Θ* for any finite p and because D(θ) > 0 by Condition NS, the verification by Lieberman et al. (2003) of Assumptions 2 and 3 of Durbin (1980) goes through without change. █

Proof of Theorem 6. The proof of the theorem relies on a passage from the result of Corollary 5 to the result of the theorem. The proof of Theorem 4 of Lieberman et al. (2003) shows that the argument of Bhattacharya and Ghosh (1978, Theorems 2 and 3) extends to the long-memory case. The main step in the proof concerns tail probability behavior of the centered WLDs, as in the equations in (2.32) of Bhattacharya and Ghosh (1978). As shown in Taniguchi and Kakizawa (2000, proof of Theorem 4.2.7), slightly weaker conditions than those of (2.32) are sufficient for the Bhattacharya and Ghosh (1978) proof of Theorem 3 to go through. These weaker conditions can be verified straightforwardly using Markov's inequality in Taniguchi and Kakizawa's case and in our case, rather than using von Bahr's inequality for i.i.d. random variables as Bhattacharya and Ghosh (1978) do. This completes the proof. █