The effect of round-off error on long memory processes

Appendix A: Distributional properties

In this appendix we consider the distributional properties of the discretization of a generic stationary Gaussian process.

From (5) the m-th moment of the discretized process can be written as E[Xdm]=∑n=−∞∞qn(nδ)m. Since X is Gaussian distributed, the variance D_d of the discretized process can be calculated explicitly. From the above expression with m=2 we obtain

(34)Dd=D∑n=−∞∞n22χ[erf(2n+122χ)−erf(2n−122χ)] (34)

Left panel of Figure 1 shows the ratio D_d/D as a function of the scaling parameter χ. It is worth noting that this ratio is not monotonic. For small χ the ratio goes to zero because δ is very large relatively to D and essentially all the probability mass falls in the bin centered at zero. In this regime the variance ratio goes to zero as

DdD22πe−1/8χχ.

When χ>>1 the ratio tends to one because the effect of discretization becomes irrelevant. In this regime D_d/D≃1+1/(12χ).

Analogously it is possible to calculate the kurtosis κd=E[Xd4]/(E[Xd2])2 of the discretized process. It is direct to show that the kurtosis is

(35)κd=∑n=1∞n4[erf(2n+122χ)−erf(2n−122χ)](∑n=1∞n2[erf(2n+122χ)−erf(2n−122χ)])2 (35)

For small χ the kurtosis diverges as

κdπ8χe1/8χ

because the fourth moment goes to zero slower than the squared second moment. For large χ the kurtosis converges as expected to the Gaussian value 3 as κ_d≃3–1/(120χ²). Note that the kurtosis reaches its asymptotic value 3 from below, since it reaches a minimum of roughly 2.982 at χ≃0.53 and then converges to three from below.

Appendix B: Proofs for Section 3

Proof of Proposition 1. The discretized process, X_d(t), is a non-linear transformation of the underlying real-valued process, X(t). Let us denote the discretization transformation with g(‧), so that X_d(t)=g(X(t)). From (10) we know that

γd(k)=∑j=1∞gj2ρj(k)  ∀ k=0, 1, …

where ρ is the autocorrelation function of the underlying continuous process, and g_j are defined as in (11).

Since the function g(x) is an odd function, g_j=0 for j even, while all the odd coefficients are non-vanishing. Therefore, the discretization function has Hermite rank 1 and can be written as an infinite sum of Hermite (odd) polynomials. The generic g_j coefficient is

gj=12πD∑n=−∞∞nδ∫(n−1/2)δ(n+1/2)δHj(xD)e−x2/2Ddx=D2π∑n=−∞∞nχ∫(n−1/2)/χ(n+1/2)/χHj(x)e−x2/2dx

The first Hermite polynomial is H₁(x)=x and the coefficient g₁ is

(36)g1D=2πχ∑k=0∞exp[−(2k+1)28χ]=12πχϑ2(0, e−1/2χ) (36)

where ϑ_a(u, q) is the elliptic theta function. For large χ, g1≃D, while for small χ the coefficient g₁ goes to zero as

g1≃D2πe−1/8χχ

The second non-vanishing coefficient g₃ is given by

(37)g3D=−13πχ∑k=0∞exp[−(2k+1)28χ]+148πχ3∑k=0∞(2k+1)2exp[−(2k+1)28χ] (37)

In principle one could calculate all the coefficients g_j. Here we want to focus on the case when the correlation coefficient ρ is small (i.e., k is large), so that it suffices to consider the first two coefficients g₁ and g₃. Therefore, from (10) we have

(38)γd(k)=g12Dγ(k)(1+O(γ(k)2))  as k→∞ . (38)

If we plug (1) and (36) into (38), we get the result.   □

Proof of Proposition 2. Under the assumption that L∈ℒ, we can write ∑i=0∞bik−βi=b0(1+O(k−β1)); by substituting this expression for L(k) into (12) we get the result.   □

Proof of Corollary 1. It follows from the definition of autocorrelation function and Proposition 1.   □

Proof of Proposition 3. From Proposition 1 we can write γd(k)~(ϑ2(0, e−1/2χ)2πχ)2k2H−2L(k), as k→∞. Then, the proposition follows from Theorem 3.3 (a) in Palma (2007).   □

Proof of Proposition 4. For the sake of simplicity, we consider the case of an underlying process with unit variance, namely D=1. In order to extend the proof to non-unit variance we need simply to do the transformation L(k)→L(k)/D. Obviously, L(k)/D is still slowly varying at infinity. Moreover, we consider only the case I=∞, the case I<∞ being trivial.

This proof is divided in two parts: in the first part we prove (17) and (18) under the assumption that L∈ℒ; in the second part we prove (19) under the additional Assumption 1.

First part From Proposition 2 and Theorem 2.1 in Beran (1994) we know that the spectral density ϕ_d(ω) exists. Moreover, if L∈ℒ, then ∃K>0 s.t. ∀ k≥K, L(k)=∑i=0∞bik−βi and the series converges absolutely. Then, from Herglotz’s theorem and (10) we can write

ϕd(ω)=Dd2π+1π∑k=1K−1γd(k)cos(kω)+1π∑k≥K∞∑j=0∞g2j+12(k2H−2∑i=0∞bik−βi)2j+1cos(kω)

As ω→0⁺, the first two terms on the RHS converge to a constant, while the third term diverges.

Since ∑ibik−βi converges absolutely ∀k≥K, we can write

∑i=0∞bik−βi=b0+b1k−βi+b2k−β2(1+o(1))

Therefore,

(39)∑k≥K∞∑j=0∞g2j+12(k2H−2∑i=0∞bik−βi)2j+1cos(kω)=∑k≥K∞{g12k2H−2[b0+b1k−β1+b2k−β2(1+o(1))] +g32k6H−6[b03+3b02b1k−β1(1+o(1))]+g52k10H−10b05(1+o(1))}cos(kω)=g12b0∑k=1∞k2H−2cos(kω)+g12b1∑k=1∞k2H−2−β1cos(kω)+O(∑k=1∞k2H−2−β2cos(kω)) +g32b03∑k=1∞k6H−6cos(kω)+O(∑k=1∞k6H−6−β1cos(kω))+O(∑k=1∞k10H−10cos(kω))+O(1) , (39)

where the term O(1) comes from substituting ∑k≥K∞ with ∑k=1∞.

Then, we introduce the following representation of a trigonometric series

(40)∑k=1∞k−scos(kω)=12(Lis(e−iω)+Lis(eiω)) (40)

where Lis(z)=∑k=1∞zk/ks is the polylogarithm. The series expansion of Li_s(e^z) for small z is (see Gradshteyn and Ryzhik 1980)

(41)Lis(ez)=Γ(1−s) (−z)s−1+∑k=0∞ζ(s−k)k! zk ,  if  s∉ℕ (41)

(42)Lis(ez)=zs−1(s−1)![Hs−1−ln(−z)]+∑k=0, k≠s−1∞ζ(s−k)k! zk ,  if  s∈ℕ (42)

where ζ(‧) is the analytic continuation of the Riemann zeta function over the complex plane and H_s is the sth harmonic number.

Finally, we plug (41) or (42) into (40), and then we apply (40) into (39). By substituting (36) for g12 and after some algebraic manipulation we get the result.

Second part Under Assumption 1 (i) the series L(k)=∑i=0∞bik−βi converges absolutely ∀k≥1. Therefore, by Mertens’ theorem,^[9] for any j≥0 we can write

(∑i=0∞bik−βi)2j+1=∑i=0∞b˜j,ik−β˜j,i  ∀ k≥1 ,

where the series on the RHS is the Cauchy product. Note that b˜0,i=bi and β˜0,i=βi ∀i, while b˜j,0=b02j+1 and β˜j,0=0 ∀j. By absolute convergence of the original series the Cauchy product also converges absolutely ∀k≥1.

From Herglotz’s theorem and Proposition 2 we can write

ϕd(ω)=Dd2π+1π∑k=1∞∑j=0∞∑i=0∞g2j+12b˜j,ik−αj,icos(kω) ,

where αj,i≡(2j+1)(2−2H)+β˜j,i.

Since {g2j+12} are bounded above (because ∑jg2j+12=Dd<∞), Assumption 1 (i) implies that the double series ∑j∑ig2j+12b˜j,ik−αj,i converges absolutely ∀k≥1. Moreover, under Assumption 1 (ii) there is only a finite number of terms in L(k) (and therefore in ∑j∑ig2j+12b˜j,ik−αj,i) that are not summable over k. Thus, from Rudin (1976) (Chapter 8, Theorem 8.3) we can invert the order of summation and write

ϕd(ω)=Dd2π+1π∑j=0∞g2j+12∑i=0∞b˜j,i∑k=1∞k−αj,icos(kω)

In other words, the Fourier transform of the series becomes the series of the Fourier transforms. From this point on the proof is very similar to that of Lemma 1 and therefore omitted for brevity.   □

Proof of Corollary 2. If L is analytic at infinity, then L(k)=∑n=0∞bnk−n for large k for some {b_n}, and the series converges absolutely within its radius of convergence. Then, L∈ℒ with β1∈ℕ, and by applying Proposition 4 and (36) we get (20). It is straightforward to see that c₂>0 and c₁>0.   □

Proof of Corollary 3. For a fGn L(k)=D2((1+1k)2H+(1−1k)2H−2)k2, which is an even analytic function at infinity and whose power series converges absolutely ∀k≥1. Because L(k) is analytic and even, β_i=2i ∀i≥0 and we can write

Γ(2H−1−βi)=(−1)2i−1Γ(1−2H)Γ(2H)Γ(2(i+1−H))→0  as i→∞ .

Thus, the autocovariance of a fGn satisfies Assumption 1, and therefore from Corollary 2 it follows that the spectral density of a discretized fGn satisfies (20) with c₀ given by (19).

From Corollary 2 we already know that the second-order term is strictly positive if H≥5/6. Hence, we just need to prove that c₀>0 if H<5/6. Since c₀ is given by (19) we can write

(43)c0=Dd2π+g12π∑i=0∞(2H2(i+1)) ζ(2(i+1−H))+∑j=1∞∑i=0∞g2j+12πD2j+1b˜j,i ζ((2j+1)(2−2H)+β˜j,i) (43)

where we used the fact that for a fGn bi=(2H2(i+1))D and β_i=2i ∀i≥0. The symbol (⋅⋅) denotes the generalized binomial coefficient.

First, since {b_i} are strictly positive and (2j+1)(2–2H)>1 ∀j≥1 if H<5/6, the third term on the RHS of (43) is strictly positive for H<5/6.

Second, from Sinai (1976) and Beran (1994) we know that the spectral density of a fGn satisfies

ϕ(ω)=2cϕ*(1−cosω)∑j=−∞∞|2πj+ω|−2H−1 , ω∈[−π, π] .

where cϕ*=D2πsin(πH)Γ(2H+1). For small ω H∈[0.5, 1] the behavior of the above spectral density follows by Taylor expansion at zero:

(44)ϕ(ω)=cϕ*|ω|1−2H−112cϕ*|ω|3−2H+o(|ω|3−2H) . (44)

On the other hand, because L(k) is analytic with β₁=2, it satisfies Assumption 2; therefore, the fGn satisfies the conditions of Lemma 1. Following the proof of that lemma, after some algebraic manipulations, we get

(45)ϕ(ω)=cϕ*|ω|1−2H+c0*−112cϕ*|ω|3−2H+o(|ω|3−2H) , (45)

where c0*=Dπ−1(12+∑i=0∞(2H2(i+1))ζ(2(i+1−H))). By comparing (44) and (45) it follows that c0* must be equal to zero, and therefore

(46)∑i=0∞(2H2(i+1))ζ(2(i+1−H))=−12 . (46)

Finally, note that from (10) we know that Dd=∑j=0∞g2j+12. Hence, by plugging (46) into (43) we obtain

c0=∑j=0∞g2j+122π−g122π+∑j=1∞∑i=0∞g2j+12πD2j+1 b˜j,i ζ((2j+1)(2−2H)+β˜j,i)=∑j=1∞g2j+122π+∑j=1∞∑i=0∞g2j+12πD2j+1 b˜j,i ζ((2j+1)(2−2H)+β˜j,i)>0

□

Appendix C: Proofs for Section 4

Proof of Lemma 1. From Theorem 2.1 in Beran (1994) we know that the spectral density ϕ(ω) exists and from Hergotz’s theorem it is the discrete Fourier transform of the autocovariance γ(k)=k2H−2∑ibik−βi.

Let α_i=2–2H+β_i ∀i≥0. Under Assumption 1 (ii) there is only a finite number of terms in the autocovariance of X that are not summable, and therefore there will be only a finite number of divergent terms in the spectral density. Moreover, under Assumption 1 (i) the series ∑i=0Ibik−βi converges absolutely ∀k≥1. Therefore, by Rudin (1976) (Theorem 8.3) we can write

(47)ϕ(ω)=D2π+1π∑i=0Ibi∑k=1∞k−αicos(kω) (47)

By using the polylogarithm representation (40) introduced above, for small ω we can plug (41) and (42) into (47). Let I1={i∈ℕ:αi∉ℕ} and I2={i∈ℕ:αi∈ℕ}. Then, under Assumption 1 (iii) we can rearrange the double series in (47) in the following way

(48)ϕ(ω)=D2π+1π b0(Γ(2H−1)sin((1−H)π)|ω|1−2H+∑j=0∞(−1)jζ(2−2H−2j)(2j)!ω2j)+1π∑i∈I1bi(Γ(1−αi)sin(παi2)|ω|αi−1+∑j=0∞(−1)jζ(αi−2j)(2j)!ω2j)+1π∑i∈I2bi(|ω|αi−1(αi−1)![sin(παi2)Hαi−1+π2cos(παi2)−sin(παi2)ln|ω|]  +∑j=0, j≠αi−1∞(−1)jζ(αi−2j)(2j)!ω2j) ,  as  ω→0+ , (48)

where ζ(‧) is the analytic continuation of the Riemann zeta function over the complex plane and H_s is the sth harmonic number.

Under Assumption 1 we can collect all the terms of the same oder and rearrange (48) in powers of ω. Let c0* be the term of order O(1) in (48); then,

c0*=D2π+1π∑i: αi≠1Ibiζ(αi) .

Under Assumption 2 α₁≠1, and therefore, if also α₁≠2, we can write

(49)ϕ(ω)=cϕb0|ω|1−2H+c0*+c1*|ω|1−2H+β1+o(|ω|min(0,1−2H+β1))  as  ω→0+ , (49)

where c_ϕ is defined as in Proposition 3 and c1*=π−1b1Γ(2H−1−β1)sin(2H−β12π).

If α₁=2, by Assumption 2 c0*≠0 and we can write

(50)ϕ(ω)=cϕb0|ω|1−2H+c0*+o(1) , as  ω→0+. (50)

Putting together (49) and (50), and noting that if c0*=0 in (49), then by Assumption 2 β₁≤2, we get the result

ϕ(ω)=cϕ|ω|1−2H(b0+cβ|ω|β+o(|ω|β) ), as  ω→0+,

where c_β≠0 and β∈(0, 2].

□

Proof of Theorem 1. Following the proof of Theorem 4 in DGH, because j₀=1 we can write Y_t as a signal-plus-noise process Y_t=W_t+Z_t, where

Wt=g1H1(Xt)=g1Xt  Zt=∑j=j1∞gjHj(Xt)

where j₁ is the second non-vanishing term in the Hermite expansion and H_j(‧) is the jth Hermite polynomial.

Part (i) If L∈ℒ, the spectral density of W_t satisfies Assumption A in DGH, i.e.,

ϕw(ω)=cw|ω|1−2H(1+o(1) ), as  ω→0+,

where cw=(g12/D)cϕb0. Moreover, since X_t is a stationary purely non-deterministic Gaussian process, it is also linear with finite fourth moments. Consequently, W_t=g₁X_t is also linear with finite fourth moments. Then, we can write

Wt=∑j=0∞ajεt−j

where ∑j=0∞aj2<∞ and ε_t are i.i.d. Gaussian variables with zero mean and unit variance. Let α(ω)=∑j=0∞ajeijω. From Proposition 4 in DGH it follows that W_t satisfies also Assumption B therein.

We show below that the spectral density of Z_t satisfies ϕz(ω)≤C|ω|1−2Hz, as ω→0⁺, for some C>0 and H_z≥0.5 such that

(51)H>Hz={0.5if  j1(2−2H)>10.5−j1(1−H)if  j1(2−2H)<1εif  j1(2−2H)=1 (51)

for any ε∈(0.5, H).

Indeed, if j₁(2–2H)>1 from (10)

∑k=1∞γz(k)=∑k=1∞∑j=j1∞gj2(ρ(k))j≤C∑k=1∞k−j1(2−2H)<∞

for some C>0. Therefore, H_z=0.5<H.

If j₁(2–2H)<1, we can prove that

ϕz(ω)=gj12Dj1b0cϕ|ω|1−2Hz(1+o(1))≤C|ω|1−2Hz  as ω→0+ ,

for some C>0 and H_z=H–(j₁–1)(1–H)<H∈(0.5, 1). Similarly, if j₁(2–2H)=1, we can prove that

ϕz(ω)=C In|ω|−1(1+o(1))≤C|ω|−ε, as ω→0+,

for some C>0 and for any ε>0. The proof of the above results is a special case of the proof of Proposition 4, and thus omitted. The results above prove (51).

Since W_t satisfies Assumptions A and B in DGH and the spectral density ϕ_z satisfies the asymptotic conditions above, consistency of H^LWY follows from Theorem 3 (i) in DGH.

Moreover, if we write the periodogram of Y_t as I_Y(ω_j)=I_W(ω_j)+v_j, where I_W is the periodogram of the “signal” W_t and v_j is the contribution of the “noise” Z_t at the jth Fourier frequency, it is straightforward to show (see DGH pp. 225–226) that

(52)H^LWY−H=(H^LWX−H)(1+oP(1))−(m−1∑j=1m(log(jm)+1)vjcwωj1−2H)(1+oP(1))+OP(logmm)=(H^LWX−H)(1+oP(1))+OP((mn)H−Hz+logmm) , (52)

where H^LWX denotes the LW estimator of {X_t} if the sequence {X_t} were observed.

Note that, roughly speaking, v_j represents the sample estimate of the higher-order terms of the spectral density of Y_t at the jth Fourier frequency. For the discretization of a fGn we know from Corollary 3 that the second-order term of the spectral density is strictly positive for all H; therefore, in that case, we expect that the second term on the RHS of the first line of (52) will induce a negative finite sample bias on H^LWY. The order of magnitude of this finite sample bias is OP((m/n)H−Hz).

Part (ii) Under Assumptions 1 and 2, from Lemma 1 it follows that X_t and therefore W_t satisfy Assumption T(α₀, β) in DGH, with α₀=2H–1 and β defined as in Lemma 1. Moreover, under Assumption 3 we can combine the second part of Proposition 5 in DGH with Proposition 3 and Theorem 2 therein, and under the assumption that m=o(m^2β/(2β+1)) we can write

(53)H^LWX−H=−(mn)β(cβb0)Bβ2−Vm2(1+oP(1))+op(m−1/2+(mn)β) (53)

where c_β is defined as in Lemma 1, B_β=(2π)^ββ/(β+1)², and

Vm=m−1∑j=1m(log(jm)+1)(ηj−Eηj)

with η_j=I_X(ω_j)/ϕ_x(ω_j).

Let r=H–H_z. By plugging (53) into (52) we obtain

(54)H^LWY−H=−Vm2−(mn)β(cβb0)Bβ2+OP((mn)r+logmm)+op(m−1/2+(mn)β+Vm) . (54)

Moreover, under Assumption 3 and m=o(m^2β/(2β+1)), by Robinson’s (1995) Theorem 2

(55)mVm→dN(0, 1) ,  as  n→∞ . (55)

Therefore, V_m=O_P(m^–1/2) and from (54) follows (26).

Part (iii) If m=o(n^2r/(2r+1)), equation (27) follows from applying (55) in (54).   □

Proof of Corollary 4. The result of the corollary follows directly from Theorem 1, and from noticing that the second non-vaninshing Hermite coefficient for the discretized process is g₃≠0, so that j₁=3.

□

For the proof of Theorem 2 we need the following lemmas. Note that the proofs of the lemmas are at the end of this Appendix.

Lemma 2Letm∈ℤ, α<1, and

(56)R(α)=−α(α−1)∫1mB2({1−t})2!tα−2dt (56)

(57)R˜(α)=−∫1mB2({1−t})2!tα−2(α(α−1)logt−1+2α)dt (57)

where B₂(x)=1/6–x+x²is the third Bernoulli polynomial and {x} represents the fractional part of the real number x. Then, both R(α) and R˜(α) converge to a finite number as m→∞.

Note that R(α) and R˜(α) are the remainders of a first order Euler-Maclaurin expansion of the sums ∑k=1mkα and ∑k=1mkαlogk, respectively.

Lemma 3Leti∈ℕ,α>0, and α≠1, 2. Then, as i→∞,

∑k=1i(i−k)k−α=A0(α)i2−α+A1(α) i+O(1)

where A0(α)=1(1−α)(2−α) and A1(α)=−(α+2)(α+3)121−α)+R(α) and R(‧) is defined as in (56).

Lemma 4Let i∈ℕ. Then, as i→∞,

∑k=1i(i−k)k−1=ilni+(−512+R1)i+O(1)∑k=1i(i−k)k−2=(53+R2)i−lni+O(1)

where R₁≡R(α=–1) and R₂≡R(α=–2).

Before proving Theorem 2 we prove the following proposition.

Proposition 5Under the assumptions of Theorem 2, let us define Σ_m=Cov(Y(i), Y(j)), i.e., the covariance matrix of the integrated process (Y(1), …, Y(m)). Then,

(i) if β≠2H–1, then

Σm=A2H(2H−1)(i2H(1+O(i−min(2H−1,β)))+j2H(1+O(j−min(2H−1,β)))+−|i−j|2H(1+O(|i−j|−min(2H−1,β)))]

(ii) if β=2H–1, then

Σm=A2H(2H−1)[i2H(1+O(i1−2Hlni))+j2H(1+O(j1−2Hlnj))+−|i−j|2H(1+O(|i−j|1−2Hln|i−j|))]

Proof. Under the assumptions on X(t), for 1≤i, j≤m we can write

(58)Σm=Cov(Y(i), Y(j))=∑k=1i∑l=1jCov(X(k), X(l))=∑k=1i(i−k) γ(k)++∑k=1j(j−k) γ(k)−∑k=1|i−j|(|i−j|−k) γ(k)+min(i, j) D (58)

where D is the variance of the process X(t). By substituting the explicit functional form for γ(k) we get

∑k=1i(i−k) γ(k)≤A∑k=1i(i−k) k2H−2+M∑k=1i(i−k)k2H−2−β

for some M>0 sufficiently large. By Lemma 3 we have A∑k=1i(i−k) k2H−2=A(i2H2H(2H−1)+O(i)).

Now, we consider the following cases:

(i) β≠2H–1. In this case we have to distinguish two cases.

If β≠2H, we can use Lemma 3 and obtain

∑k=1i(i−k) k2H−2−β=A0(2H−2−β) i2H−β+A1(2H−2−β) i+O(1)=i2H O(i−min(2H−1,β))

If β=2H, then we can use Lemma 4

∑k=1i(i−k) k2H−2−β=(53+R2)i−lni+O(1)=O(i)

Then, we repeat the same calculation for the second and third term in (58). By noting that min(i, j) is either of order O(i) or O(j) and putting together all the terms, we obtain the result.

(ii) β=2H–1. In this case we can use Lemma 4

∑k=1i(i−k) k2H−2−β=(ilni+(−512+R1) i+O(1))=i2HO(i1−2Hlni)

Then, we repeat the same calculation for the second and third term in (58). By noting that min(i, j) is either of order O(i) or O(j) and putting together all the terms, we obtain the result.   □

Proof of Theorem 2. First, for j∈{1, …, [n/m]} let us define the vector:

Y(j)=(Y(1+m(j−1)),…,Y(mj))⊤

where x^⊤ means the transpose of x.

Then, following Bardet and Kammoun (2008),

F12(m)=1m(Y(1)−PE1Y(1))⊤(Y(1)−PE1Y(1))=1m(Y(1)⊤Y(1)−Y(1)⊤PE1Y(1)),

where E₁ is the vector subspace of ℝm generated by the two vectors e₁=(1, …, 1)^⊤ and e₂=(1, 2, …, m)^⊤, PE1 is the matrix of the orthogonal projection on E₁, and the second equality holds because the projection operator is idempotent. As a consequence,

E[F12(m)]=1m(Tr(Σm)−Tr(PE1Σm))

where Tr(·) is the trace of a square matrix.

Case (i) If β≠2H–1, from Proposition 5 we get

Tr(Σm)=AH(2H−1)m2H+1(∫01x2Hdx+O(m−min(2H−1,β))+O(m−1)),

where the error O(m^–1) comes from approximating the sum with the integral; therefore,

(59)Tr(Σm)=Am2H+1H(2H−1)12H+1(1+O(m−min(2H−1,β))) (59)

For the term Tr(PE1Σm), we use the following representation of the projection operator

(60)PE1=2m(m−1)((2m+1)−3(i+j)+6i⋅j1+m). (60)

Then, using (60) and Proposition 5 we can write

Tr(PE1Σm)=2Am2H+1m2m(m−1)2H(2H−1)[∑i=1m∑j=1m1m2((2+1m)−3i+jm+6ijm(m+1))⋅ ((im)2H(1+O(i−min(2H−1,β)))+(jm)2H(1+O(j−min(2H−1,β)))+ −(|i−j|m)2H(1+O(|i−j|−min(2H−1,β))))].

Approximating sums with integrals we get

(61)Tr(PE1Σm)=Am2H+1H(2H−1)(1+O(m−min(2H−1,β))+O(m−1))⋅ ⋅∫01∫01[(2−3(x+y)+6xy)(x2H+y2H−|x−y|2H)]dxdy=Am2H+1H(2H−1)1+H(4+H)(1+H)(2+H)(1+2H)(1+O(m−min(2H−1,β))) (61)

Putting together (59) and (61) we obtain

1m(Tr(Σm)−Tr(PE1Σm))=AH(2H−1)f(H)m2H(1+O(m−min(2H−1,β)))

which is the formula of E[F12(m)].

Case (ii) If β≠2H–1, the proof is exactly the same, except for replacing all the terms O(i^{–min (2H–1,β)}) with the terms O(i^1–2Hln i).   □

Proof of Corollary 5. It follows from the autocovariance of a fractional Gaussian noise (see formula (3)). The proof is very similar to the proof of Theorem 2, and thus omitted. However, a complete proof can be found in Bardet and Kammoun (2008) (see Proof of Property 3.1 therein).   □

Proof of Corollary 6. It follows from Proposition 2 and Theorem 2.   □