2.1. Process Assumptions

We suppose that the observed data

represent a realization from a stationary, real-valued LRD process

that satisfies the following assumption.

Assumption L. For independent identically distributed (i.i.d.) innovations

with mean E(ε_t) = 0 and 0 < E(ε_t²) < ∞, it holds that

where

and the real sequence

is square summable

such that the autocovariance function r(k) = Cov(Y_t,Y_t+k) admits a representation as in (1).

Assumption L encompasses two popular models for strong dependence: the fractional Gaussian processes of Mandelbrot and van Ness (1968) and the fractional autoregressive integrated moving average (FARIMA) models of Adenstedt (1974), Granger and Joyeux (1980), and Hosking (1981). For FARIMA processes in particular, we permit greater distributional flexibility through possibly non-Gaussian innovations. Note that a LRD FARIMA(0,d,0) series, d ∈ (0,½), admits a casual moving average representation

involving the gamma function Γ(·). More general FARIMA series, for which (1) holds with α = 1 − 2d and constant L₁(·) = C₁ > 0, follow from applying an autoregressive moving average (ARMA) filter to a process from (2) (cf. Beran, 1994).

We remark that the results of this paper also hold by stipulating the long-range dependence through regularity conditions, as in Theorem 2.2 of Hall et al. (1998), on the spectral density f of the process {Y_t} for which lim_x→0 f (x)/{|x|^α−1L₁(1/|x|)} > 0 exists finitely. Under certain conditions, this behavior of f at the origin is equivalent to (1) and serves as an alternative description of long-range dependence (Bingham, Goldie, and Teugels, 1987). However, our assumptions here on the LRD linear process are fairly mild, requiring i.i.d. innovations to only have a bounded second moment.

2.2. Distributional Properties of the Sample Mean

In the following discussion, for any two nonzero real sequences {s_n} and {t_n}, we write s_n ∼ t_n if lim_n→∞ s_n /t_n = 1. With the proper scaling d_n, the asymptotic distribution of the normalized sample mean is known to be normal for Assumption L processes (cf. Davydov, 1970): as n → ∞,

where Z represents a standard normal variable and

denotes convergence in distribution. However, setting confidence intervals for μ, based on Y_n and its large-sample normal distribution, becomes complicated for linear LRD processes. The covariance decay rate in (1) implies that the variance of Y_n converges to 0 as follows:

for L(·) = 2{(2 − α)(1 − α)}⁻¹L₁(·), which is slower than the usual O(n⁻¹) rate associated with weakly dependent data. Consequently, the correct scaling d_n = {n^2−αL(n)}^1/2 ∼ {Var(nY_n)}^1/2 for n(Y_n − μ)/d_n to have a normal limit depends on the unknown quantities α,L(·) from (4).

With additional assumptions on the linear process, unknown quantities in d_n could in principle be estimated directly for interval estimation of μ based on a normal approximation (3). For example, by assuming a constant function L₁(·) = C₁ in (1) (along with additional regularity conditions on f), estimates

of α,C₁ could be obtained through various periodogram-based techniques (Bardet, Lang, Oppenheim, Philippe, Stoev, and Taqqu, 2003). However, after substituting such estimates directly into d_n from (3), the resulting Studentized mean

may fail to adequately follow a normal distribution. To illustrate, we conducted a small numerical study of the coverage accuracy of confidence bounds for the mean μ of several LRD FARIMA processes, set with a normal approximation for a Studentized mean G_n. For these processes, (1) holds with a function L₁(·) = C₁. We obtained a version G_n by estimating α and C₁ in d_n through popular log-periodogram regression (Geweke and Porter-Hudak, 1983) using the first

Fourier frequencies (Hurvich, Deo, and Brodsky, 1998). The coverage probabilities in Table 1 suggest that the normal distribution may not always appropriately describe the behavior of a Studentized sample mean obtained through such plug-in estimates in (3). (The LRD processes in Table 1 involve filtered FARIMA(0,d,0) series, but other simulation results indicate that a plug-in version G_n may produce better confidence intervals with unfiltered FARIMA(0,d,0) series.) We remark that, with Gaussian LRD processes, Beran (1989) developed a modified normal distribution for Y_n after Studentization with a periodogram-based estimate of α. However, the approach given was globally parametric in requiring the form of the spectral density f to be known on the entire interval [0,π], which is a strong assumption.

Coverage probabilities for one-sided 90% lower and upper confidence bounds

In Section 3, we show that the sampling window method produces consistent, nonparametric estimates of the finite-sample distribution of the sample mean from strongly dependent linear processes. Subsampling distribution estimators for Y_n can then be used to calibrate nonparametric confidence intervals for the process mean μ. Under linear long-range dependence, an advantage of this approach over traditional large-sample theory is that the subsampling confidence intervals may be constructed without making restrictive assumptions on the behavior of f near zero and without estimating the covariance parameter α. Another benefit of the subsampling method is its applicability to other formulations of long-range dependence involving nonlinear processes. That is, the subsampling technique has established validity with transformed-Gaussian LRD series as treated in Hall et al. (1998). For these series, n(Y_n − μ)/d_n may have a nonnormal limit distribution, and a normal approximation for the sample mean might break down.

6.1. Proofs of Main Results

In the following discussion, let σ_n² = n² Var(Y_n). Denote the supremum norm

for a function

and let Φ denote the standard normal distribution function. Unless otherwise specified, limits in order symbols are taken letting n → ∞.

We first state a useful result concerning moments of the sample mean Y_n from a LRD linear process. Lemma 1(a) follows from the proof of Theorem 18.6.5 in Ibragimov and Linnik (1971) and bounds sums of consecutive filter coefficients in terms of the standard deviation of nY_n; part (b) of Lemma 1 corresponds to Lemma 4 of Davydov (1970).

LEMMA 1. Suppose Assumption L holds. For all

,

(a) and for all

,

(b) E{[n(Y_n − μ)]^2k} ≤ A_k(σ_n²)^k for some A_k > 0, if E(|ε₀|^2k) < ∞ for a given

.

For the proof of Theorem 1, we define

. Note that

differs from the sampling window estimator

from Section 3.1 by centering subsample sums with [ell ]μ rather than [ell ]Y_n. We also require the following result for LRD linear processes with bounded innovations. For these series, Lemma 2(a) shows that standardized subsample sums based on well-separated blocks are asymptotically uncorrelated, whereas Lemma 2(b) establishes the convergence of

. We defer the proof of Lemma 2 to Section 6.2.

LEMMA 2. Suppose the conditions of Theorem 1 hold with bounded innovations, that is, P(|ε_t| ≤ B) = 1 for some B > 0. Then, as n → ∞,

where

is a standard normal variable. For a > 0, E(Z^a) = (a − 1)(a − 3)…(1) for even a; 0 otherwise.

(b)

.

Proof of Theorem 1. We note that (4) and the assumption [ell ]⁻¹ + n^−1+δ[ell ] = o(1) imply

because L is positive and x^γL(x) → ∞, x^−γL(x) → 0 as x → ∞ for any γ > 0 (Ibragimov and Linnik, 1971, App. 1). We can bound

and

for each ε > 0. From (7), we find P([ell ]|Y_n − μ|/d_{[ell ]} > ε) = o(1) by Chebychev's inequality using σ_n² ∼ d_n² from (4). From the continuity of Φ, it follows that ∥F_n − Φ∥_∞ = o(1) by (3) and also that

as ε → 0. Hence, it suffices to show

or, equivalently as a result of Φ's continuity,

for each

. We will prove

for

.

Let E(ε_t²) = τ² > 0. For each

, define variables

, where τ_b² = E(ε_0,b²). (We may assume τ_b² > 0 w.l.o.g. in the following discussion.) For each

, write

to denote the analogs of

with respect to Y_1,b,…,Y_n,b. Note that both series {Y_t} and {Y_t,b} involve the same linear filter with i.i.d. innovations of mean 0 and variance τ² and hence have the same covariances; in particular, we may set d_n,b² = d_n² for

. For any

,

Letting D_{[ell ],b} = (S_{[ell ]1,b} − S_{[ell ]1})/d_{[ell ]},

where P(|D_{[ell ],b}| > ε) ≤ Var(S_{[ell ]1} − S_{[ell ]1,b})/(ε²d_{[ell ]}²), and we deduce

by the i.i.d. property of innovations. Hence, for any

,

using (4), A_2n,b(x) = o(1) as n → ∞ by Lemma 2(b), and |F_{[ell ]}(x) − Φ(x)|,∥F_{[ell ],b} − Φ∥_∞ = o(1) as n → ∞ by (3). Because

as ε → 0, the proof of Theorem 1 is finished. █

Proof of Theorem 2. Let m denote m_kn, k ∈ {1,2}, and define

for N_m = n − m + 1. Using Hölder's inequality, (4), and (7), we can show that

With the truncated variables from the proof of Theorem 1, let

with

; again the processes {Y_t} and {Y_t,b} have the same covariances for all

. In a fashion similar to (8), we find by Hölder's inequality,

using σ_m² = Var(S_m1) = Var(S_m1,b) and Var(S_m1 − S_m1,b) = 2σ_m²{1 − τ_b⁻¹τ⁻¹E(ε₀ε_0,b)} from before.

Applying Lemma 2(a) and the bound on E{(S_m1,b − mμ)⁴} from Lemma 1(b) with (4),

holds for each

. Then

follows from using (8)–(10) to deduce

and then applying lim_b→∞ τ_b⁻¹E(ε₀ε_0,b) = τ. Because

and (5) and (6) imply

the convergence

now follows. From this and Theorem 1, we find the convergence of

in probability as in Hall et al. (1998). █

Proof of Theorem 3. From Corollary 6.1.1.2 of Fuller (1996), we have

. Lemmas 1 and 2 and the same proofs of Theorems 1 and 2 (including Lemma 2) apply with the convention that α = 1,

under short-range dependence. The one modification is that (7) still holds if [ell ]⁻¹ + [ell ]/n = o(1). █

6.2. Proof of Technical Lemma

Proof of Lemma 2(b). The result follows from Theorem 2.4 of Hall et al. (1998) and its proof after verifying that the conditions required are met: the process {Y_t} has all moments finite by assumption; n(Y_n − μ)/σ_n converges to a (normal) continuous distribution by (3) which is uniquely determined by its moments; ([ell ]²σ_n²)/(n²σ_{[ell ]}²) = o(1) by (4) and (7); and Lemma 2(a) holds. █

Proof of Lemma 2(a). It suffices to consider only positive

, because Lemma 1(b) with (3) implies that E[(S_{[ell ]1}*)^a] → E(Z^a) as n → ∞ for any nonnegative a.

We establish some additional notation. For

, write the standardized subsample sum

using

and set E(ε_t²) = 1 throughout the proof because of standardization; we suppress here the dependence of d_k(i) on [ell ] in our notation. Write the nonnegative integers as

. For

, denote integer vectors

; write a set

and define with

,

where indices in the sum

extend over integer m-tuples

with distinct components k_j ≠ k_j′ for 1 ≤ j ≠ j′ ≤ m. We will later use that

holds for

so that certain sums Δ_i,a,b(s^(m),t^(m)) are finitely defined. We omit the proof of (11) here (in light of showing (14) to follow, which uses similar arguments).

Because the innovations are i.i.d. with E(ε_t) = 0, it holds that

for

and integers

unless, for each 1 ≤ j ≤ a + b, there exists some j′ such that k_j = k_j′, implying less than [lfloor ](a + b)/2[rfloor ] distinct integer values among (k₁,…,k_a+b). Using this and

for any

, we rewrite (12) as a sum over collections of integer indices (k₁,…,k_a+b) with 1 ≤ m ≤ [lfloor ](a + b)/2[rfloor ] distinct values:

where the sum [sum ]_(W1,…,Wm) is taken over all size m partitions (W₁,…,W_m) of

. The indicator function I {·} in the bracketed sum in the preceding expression signifies that, for a given partition (W₁,…,W_m), we sum terms in (12) over integer indices

satisfying k_j = k_j′ if and only if j,j′ ∈ W_h, h = 1,…,m. We can more concisely write (12) as

in terms of vectors

as a function of a partition (W₁,…,W_m). By the nature of the partitions (W₁,…,W_m), the vectors (s^(m),t^(m)) in (13) are elements of B_a,b,m for some 1 ≤ m ≤ [lfloor ](a + b)/2[rfloor ] (any set W_h in a partition (W₁,…,W_m) has at least two elements by the restriction m ≤ [lfloor ](a + b)/2[rfloor ] so that min_1≤j≤m(s_j + t_j) ≥ 2 follows).

To help identify the most important terms in the summand (13), we define a count

as a function of

and also a special indicator function

. Now to show Lemma 2(a), it suffices to establish for any

that

If either a or b is odd, so that E(Z^a)E(Z^b) = 0, Lemma 2(a) follows immediately from (13) and (14). In the case that a,b are even, we find that the dominant component in (13) involves the sum over partitions (W₁,…,W_m) with m = (a + b)/2 and corresponding (s^(m),t^(m)) that satisfy

or equivalently s_j,t_j ∈ {0,2}, s_t + t_j = 2 for 1 ≤ j ≤ m = (a + b)/2; in this instance, Ψ_a,b(s^(m),t^(m)) = 1 (by E(ε₀²) = 1) and a size m = (a + b)/2 partition (W₁,…,W_(a+b)/2) of {1,…,a + b} is formed by a size a/2 a partition of {1,…,a} and a size b/2 partition of {a + 1,…,b + a}; there are exactly (a − 1)(a − 3)…(1) × (b − 1)(b − 3)…(1) such partitions possible. So for even a,b with m = (a + b)/2, it holds that

which with (13) and (14) implies that Lemma 2(a) follows for a,b even.

We now focus on proving (14) by treating two cases:

. For

, define subsample sum covariances

. Although we cannot assume that

under long-range dependence, it holds that

by applying Lemma 1(a). Similarly, if

with min_1≤j≤m max{s_j,t_j} ≥ 2, then

follows from (15), where ω_{[ell ]} = o(1).

Case 1.

. We show that (14) holds with an induction argument on m. Consider first the possibility m = 1, for which (s^(m),t^(m)) = (s₁,t₁) = (a,b). If

, then a = b = 1; in this case,

. For 0 < ε < 1 and large n such that nε/2 > [ell ], the growth rate in (1) with (4) gives

defining M_n,ε = sup{L(tn) : ε/2 < t < 2} and using (7) with M_n,ε /L(n) → 1 by Taqqu (1977, Lem. A1). If

, then either a = s₁ > 1 or b = t₁ > 1 for

, and (14) follows from (16) and O(ω_{[ell ]}^a+b−2m) = o(1).

Now assume that, for some

, (14) holds whenever

with

and m ≤ min{T,[lfloor ](a + b)/2[rfloor ]}. We let

with

in the following discussion and show that (14) must hold by the induction assumption.

If

, then min_1≤j≤m max{s_j,t_j} ≥ 2 holds and (14) follows from (16) and O(ω_{[ell ]}^a+b−2m) = o(1) because a + b > 2m, which we verify. If a,b are even, then m < (a + b)/2 must hold by

; if a or b is odd, then a + b > 2m follows because (s^(m),t^(m)) ∈ B_a,b,m and max_1≤j≤m{s_j + t_j} > 2 must hold (the alternative by (s^(m),t^(m)) ∈ B_a,b,m with

is that s_j,t_j ∈ {0,2} for 1 ≤ j ≤ m, so that

are even, a contradiction).

Consider now the possibility that

. Because m = T + 1 > 1 with a,b > 1 necessarily, say (w.l.o.g.) that components s_m = t_m = 1 in s^(m) = (s₁,…,s_m), t^(m) = (t₁,…,t_m). Using

, we can algebraically rewrite the sum

where (s₀^(m−1),t₀^(m−1)) ∈ B_{a−1,b−1,m−1} for s₀^(m−1) = (s₁,…,s_m−1), t₀^(m−1) = (t₁,…,t_m−1), and (s_j^(m−1),t_j^(m−1)) ∈ B_a,b,m−1 for s_j^(m−1) = s₀^(m−1) + e_j^(m−1), t_j^(m−1) = t₀^(m−1) + e_j^(m−1), writing the jth coordinate vector

. Note that

because m − 1 < (a + b)/2. By the induction assumption on terms Δ_i,a,b(s_j^(m−1),t_j^(m−1)), j > 0, in (18) along with (11) and (17), we find that (14) holds whenever

with m = T + 1. This completes the induction proof and the treatment of Case 1.

Case 2.

. Here a,b are even, and the components of (s^(m),t^(m)) satisfy s_j ∈ {0,2}, t_j = 2 − s_j for 1 ≤ j ≤ m = (a + b)/2. By (15),

follows for any

. Using this and algebra similar to (18), we can iteratively write Δ_i,a,b(s^(m),t^(m)) as a sum by parts equal to

where 1 ≤ h ≤ m − 1 = (a + b − 2)/2. With an argument as in (16), we find a uniform bound |R_h,i,a,b(s^(m),t^(m))| ≤ (a + b)ω_{[ell ]}²/2 = o(1) for each

, 1 ≤ h ≤ m − 1. Hence, (14) follows for Case 2. The proof of Lemma 2(a) is now complete. █