Generalized seasonal tapered block bootstrap

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Generalized seasonal tapered block bootstrap

Anna E. Dudek, Efstathios Paparoditis, Dimitris Politis

To cite this version:

Anna E. Dudek, Efstathios Paparoditis, Dimitris Politis. Generalized seasonal tapered block boot- strap. Statistics and Probability Letters, Elsevier, 2016, 115 (27-35), �10.1016/j.spl.2016.03.022�. �hal- 01310921�

Generalized Seasonal Tapered Block Bootstrap

Anna E. Dudek^1,2

1 Institut de Recherche Math´ematique de Rennes, Universit´e Rennes 2, Rennes, France,

2 AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland,

email: [email protected] Efstathios Paparoditis

Dept. of Mathematics and Statistics, University of Cyprus, Nicosia CY 1678, Cyprus

Dimitris N. Politis

Dept. of Mathematics, University of California—San Diego, La Jolla, CA 92093-0112, USA

Abstract

In this paper a new block bootstrap method for periodic times se- ries called Generalized Seasonal Tapered Block Bootstrap (GSTBB) is introduced. Consistency of the GSTBB for parameters associated with periodically correlated time series is shown; these are the overall mean, seasonal means and Fourier coefficients of the autocovariance function. Consequently, the construction of bootstrap pointwise and simultaneous confidence intervals for such parameters is possible. A simulation data example is also presented.

Keywords: autocovariance function, seasonal means, confidence inter- vals.

1 Introduction and problem formulation

In this paper we consider a modification of the Generalized Seasonal Block Bootstrap (GSBB) proposed by Dudek et al. (2014a) [3] by incorporating the tapering idea of Paparoditis and Politis (2001) [12].

First we introduce some notation. Let {X_t, t∈ Z} be a periodically corre- lated (PC) time series with known period d, i.e. X_t has periodic mean and

covariance functions

E (Xt+d) = E (Xt) and Cov (Xt+d, Xs+d) = Cov (Xt, Xs).

For more details on PC time series we refer the reader to Hurd and Miamee (2007) [8].

We will assume thatX_t is α-mixing i.e. α_X(k)→0 ask→ ∞,where α_X(k) = sup

t

sup

A∈FX(−∞,t) B∈FX(t+k,∞)

|P(A∩B)−P(A)P(B)|

and FX(−∞, t) =σ({X_s:s≤t}),FX(t+k,∞) =σ({X_s:s≥t+k}).

We observe the sample X₁, . . . , X_n. Denote the sequence of data-tapering windows byw_n(t), wherew_n(t) =w(_t−0.5

n

); here,w is a function such that w:R →[0,1] with||wn||1=∑_n

i=1|wi(t)|and ||wn||2 =(∑_n

i=1w_i²(t))_1/2 . Following Paparoditis and Politis (2001) [12] we assume the following con- ditions:

A1 w(t) = 0 if t /∈[0,1], andw(t)>0 for tin a neighbourhood of 1/2;

A2 w(t) is symmetric about t= 1/2 and nondecreasing fort∈[0,1/2];

A3 The self-convolutionw∗w(t) is twice continuously differentiable at the pointt= 0 wherew∗w(t) =∫₁

−1w(x)w(x+|t|)dx.

Define the overall mean ¯µ = d⁻¹∑_d

i=1µ_i and the seasonal means µ_i = E(Xi+jd), i= 1, . . . , d,j∈ Z that are estimated respectively by

bµ=d⁻¹

∑d i=1

b

µ_i and µb_i=v⁻_i ¹

v∑i−1 j=0

X_i+jd (1)

wherev_i = max{j such thati+jd≤n}.

We now introduce the Generalized Seasonal Tapered Block Bootstrap (GSTBB);

for simplicity, we assume thatn=lband n=wd, wherebis a block length and l, b∈ N.

BOOTSTRAP ALGORITHM (GSTBB):

• defineXe_t=X_t−µb_<t>, where < t >= (t mod d) denotes the season associated witht.

• the bootstrap sampleX₁^∗, . . . , X_n^∗ is generated by applying the GSBB procedure of Dudek et al. (2014a) to the sampleXe1, . . . ,Xen. For the sake of simplicity of notation and presentation we describe below the circular version of the GSBB. However, the usual GSBB can also be used.

– Choose a (positive) integer block sizeb(< n).

– Fort= 1, b+ 1,2b+ 1, . . . ,(l−1)b+ 1, let

B_t^∗ = (X_t^∗, X_t+1^∗ , . . . , X_t+b^∗ ₋₁) = (Xe_kt,Xe_kt+1, . . . ,Xe_kt+b−1), wherek_t is iid from a discrete uniform distribution

P(k_t=t+vd) = 1

w for v= 0,1, . . . , w−1.

When the time indext+vd > n, we shift it taking t+vd−n.

– Join thelblocksB^∗₁, . . . , B_(l^∗_−1)b+1 to obtain a bootstrap sample (X₁^∗, . . . , X_n^∗).

• form= 0, . . . , l−1, letY_mb+j^∗ :=wb(j)

√b

||w_b||2X_mb+j^∗ . GSTBB bootstrap versions ofµb and bµi are now defined by

bµ^∗=d⁻¹

∑d i=1

b

µ^∗_i and µb^∗_i =v_i⁻¹

v∑i−1 j=0

Y_i+jd^∗ . (2) Consistency of the GSTBB for the sample means and related statistics is shown in the sequel; all proofs are in the Appendix. A small simulation study can be found in Section 4.

2 Main results

Let µ = (µ₁, . . . , µ_d)^′ and µb = (µb₁, . . . ,µb_d)^′ denote the vector of seasonal means and its estimator as defined in equation (1), respectively. LetL(√

n(µb− µ)) denote the probability law of√

n(µb−µ), andL^∗(√

n(µb^∗−E^∗bµ^∗)) its boot- strap counterpart conditionally on the observed time seriesX₁, X₂, . . . , X_n; similarly, defineL(√

n(bµ−µ)), and L^∗(√

n(µb^∗−E^∗bµ^∗)).

Theorem 2.1 Let {X_t, t∈ Z} be a PC time series that is α–mixing. As- sume that for some δ >0, sup_tE|X_t|^4+δ <∞ and ∑_∞

k=1kα^δ/(4+δ)_X (k)<∞.

If b→ ∞ as n→ ∞ such that b=o(n), then the GSTBB is consistent for the overall mean and seasonal means, i.e.

d2

(L(√

v(µb−µ))

,L^∗(√

v(µb^∗−E^∗µb^∗))) p

−→0, (3)

d₂

(L(√ n

(bµ−µ ))

,L^∗(√ n

(bµ^∗−E^∗bµ^∗

))) _p

−→0, (4) where d₂ is the Mallows metric and v=⌊n/d⌋.

Furthermore, we present consistency theorems for smooth functions of the overall mean and the seasonal means; the latter is important as it allows for construction of simultaneous confidence intervals forµ.b

Theorem 2.2 Let {Xt, t∈ Z} be a PC time series that fulfills the assump- tions of Theorem 2.1. Suppose that function H:R → R is:

(i) differentiable in a neighborhood of µ

NH ={x∈ R:|x−µ|<2η} for some η >0 (ii) H^′(µ)̸= 0

(iii) the first-order derivative H^′ satisfies a Lipschitz condition of order κ >0 onN_H.

If b→ ∞ as n → ∞ such that b = o(n/logn) and b⁻¹ = o(log⁻¹n), then GSTBB is consistent i.e.

d₂

(L(√ n

( H

(bµ

)−H(µ) ))

,L^∗(√ n

( H

(bµ^∗ )−H

(

E^∗bµ^∗)))−→^p 0.

Theorem 2.3 Let {Xt, t∈ Z} be a PC time series that fulfills the assump- tions of Theorem 2.1. Suppose that function H:R^d→ R^s is:

(i) differentiable in a neighborhood N_H =

{

x∈ R^d:||x−µ||<2η }

for some η >0 (ii) ▽H(µ)̸= 0

(iii) the first-order partial derivatives of H satisfy a Lipschitz condition of order κ >0 onN_H.

If b→ ∞ as n → ∞ such that b = o(n/logn) and b⁻¹ = o(log⁻¹n), then GSTBB is consistent i.e.

d₂( L(√

v(H(bµ)−H(µ)))

,L^∗(√

v(H(µb^∗)−H(E^∗µb^∗)))) p

−→0,(5) where µb= (µb1, . . . ,µbd) andv=⌊n/d⌋.

Remark: In practical applications, the GSTBB should not be used with b ≤ d such that d = kb for k ∈ N especially for simultaneous confidence intervals. In such a case, the GSTBB provides too high or too low cover- age probabilities. To explain this phenomenon, consider the simple case of b=d. In this situation observations from the first and the last season are used with lower weights. By contrast, if b= 2d, then lower weights are no longer assigned to all observations from aforementioned seasons, and this negative effect disappears.

Remark: Using Theorem 2.3 one may calculate quantiles of the (1-2α)%

equal-tailed bootstrap simultaneous confidence intervals using the maximum and the minimum statistics. Define

K_max(x) = P^∗ (√

wmax

i (bµ^∗_i −bµ_i)≤x )

, Kmin(x) = P^∗

(√ wmin

i (bµ^∗_i −µbi)≤x )

. Then, the confidence region is of the form

( b

µ_i−K_max⁻¹ (1−α)

√w ,µb_i−K_min⁻¹ (α)

√w )

(6) simultaneously fori= 1, . . . , d.

3 Application of the GSTBB in second order mo- ment analysis

The statistical analysis of PC time series in often performed in the frequency domain. To detect significant frequencies the Fourier representations of the mean and the autocovariance functions are used. Since it is easy to demean a PC time series by removing periodic means, the main interest of researchers is focused on the autocovariance function. Thus, from here on we will assume that EX_t≡0.

Denote the autocovariance function byB(t, τ) = Cov(Xt, Xt+τ), wheretand τ are time and shift indices, respectively. Note that for a PC time series, B(t, τ) is a periodic function of t. The Fourier representation of B(t, τ) is of the form

B(t, τ) = ∑

λ∈Λτ

a(λ, τ) exp(iλt),

where Λ_τ ={λ:a(λ, τ)̸= 0} ⊂ {2kπ/d, k = 0, . . . , d−1}.Thus, the number of second order significant frequencies is finite.

Without loss of generality we assume that τ ≥0 from now on. Then, the estimator of a(λ, τ) is of the form (see Hurd (1989, 1991) [5] [6], Hurd and Le´skow (1992) [7])

b

an(λ, τ) = 1 n

n∑−τ t=1

XtXt+τexp(−iλt).

The estimator ban(λ, τ) is asymptotically normal; see Lenart et al. (2008) [10]. However, the asymptotic covariance matrix is very difficult to estimate (see Dudek et al. (2014b) [4]) and hence to construct confidence intervals resampling methods are used. For PC processes validity of a few methods for a(λ, τ) was already shown. The first consistency result was obtained by Lenart et al. [10] for subsampling method. The GSBB consistency was obtained in Dudek et al. (2014b) [4]. Finally, in Dudek (2015) [2] the Cir- cular Block Bootstrap was applied for almost periodically correlated (APC) processes. APC time series are a more general class than PC time series.

Their mean and autocovariance functions are almost periodic. In what fol- lows, we recall the GSBB estimator of a(λ, τ) of Dudek et al. (2014b) [4], we construct its GSTBB analog and show its consistency.

For a fixedτ ≥0 and λ∈Λ_τ the GSBB version ofba_n(λ, τ) is of the form b

a^GSBB_n (λ, τ) = 1 n

n∑−τ t=1

X_t^∗X_t+τ^∗ exp(−iλt).

To apply the GSTBB for a(λ, τ) we need to modify the GSTBB algorithm presented in the previous section. Without loss of generality, assume that n=vd, v ∈ Z.Note that the estimatorban(λ, τ) can be rewritten as follows:

b

an(λ, τ) = 1 vd

∑d s=1

v−1∑

k=0

X_s+kdX_s+kd+τexp(−iλ(s+kd)) = 1 d

∑d s=1

ban,s(λ, τ);

In the above, ifs+kd+τ > n we set the corresponding summand to 0.

Note that forλ∈Λ_τ we have ba_n,s(λ, τ) = 1

v

v−1

∑

k=0

X_s+kdX_s+kd+τexp(−iλ(s+kd)) = 1 v

v−1

∑

k=0

X_s+kdX_s+kd+τexp(−iλs) and finally

ba_n(λ, τ) = 1 vd

∑d s=1

exp(−iλs)

v−1

∑

k=0

X_s+kdX_s+kd+τ = 1 d

∑d s=1

exp(−iλs)˘a_n,s(λ, τ), where ˘a_n,s(λ, τ) = ¹_v∑_v−1

k=0X_s+kdX_s+kd+τ.

The estimators ban,s(λ, τ) will be essential to define the GSTBB estimator.

They will have the same role as the estimators of seasonal means in the previous section, i.e. they will be used to demean the corresponding series.

BOOTSTRAP ALGORITHM (GSTBB) fora(λ, τ) :

Letn=lb,whereb is a block length; recall the assumption EX_t≡0.

• the bootstrap sampleX₁^∗, . . . , X_n^∗is generated using GSBB onX₁, . . . , X_n;

• fors= 1, . . . , d andk= 0, . . . , v−1 let Y˘_s+kd^∗ = exp(−iλs)(

X_s+kd^∗ X_s+kd+τ^∗ −˘a_n,s(λ, τ))

;

• form= 0, . . . , l−1, letY_mb+j^∗ :=w_b(j)_||w^√b

b||2

Y˘_mb+j^∗ ;

• the GSTBB estimator is of the form b

a^∗_n(λ, τ) = 1 n

∑d s=1

v−1

∑

k=0

Y_s+kd^∗ = 1 n

n∑−τ t=1

Y_t^∗.

Before showing the consistency of the proposed algorithm, we introduce some additional notation. Let λ and τ denote r-dimensional vectors of frequencies and shifts of the form λ = (λ₁, . . . , λ_r)^′, τ = (τ₁, . . . , τ_r)^′. Additionally,

a(λ,τ) = (ℜ(a(λ₁, τ₁)),ℑ(a(λ₁, τ₁)), . . . ,ℜ(a(λ_r, τ_r)),ℑ(a(λ_r, τ_r)))^′. By ba_n(λ,τ) we denote its estimator and by ba^∗_n(λ,τ) its bootstrap coun- terpart. Additionally, by W P(k) we denote a weakly periodic process of order k. Recall that a process X_t is W P(k) if E|X_t|^k < ∞ and for any t, τ₁, . . . , τ_k−1 ∈ Z E(X_tX_t+τ1. . . X_t+τk−1) is periodic in the variable t.

Theorem 3.1 Let {Xt, t ∈ Z} be a PC time series with E(Xt) ≡ 0 and WP(4). Assume that for someδ >0,sup_tE|Xt|^8+2δ <∞and∑_∞

k=1kα^δ/(4+δ)_X (k)<

∞. Ifb→ ∞ asn→ ∞ such thatb=o(n) then the GSTBB for the overall mean is consistent, i.e.

d₂( L(√

n(ba_n(λ,τ)−a(λ,τ)))

,L^∗(√

n(ba^∗_n(λ,τ)−E^∗ba^∗_n(λ,τ)))) p

−→0.

Theorem 3.1 states consistency of the GSTBB under the same conditions that were used to show consistency of the GSBB in Dudek et al. (2014b) [4].

Remark: The consistency of the GSTBB for smooth functions ofa_n(λ,τ) can be easily obtained using the same reasoning as in Dudek et al. (2014b) [4]. Thus, we omit technical details.

4 Simulation data example

In this section we compare the performance of the GSBB and the GSTBB on a simulation data example. For our study we chose an ARMA type model of the form

X_t= 2cos(2πt/d)+0.2X_t−1+0.1X_t−2+ϵ_t+0.25cos(2πt/d) (ε_t+0.3ε_t−1+0.1ε_t−2), whereϵtare independent standard normal random variables andεtare inde- pendent random variables from normal distribution with mean 0 and stan- dard deviation 0.5. Using both bootstrap approaches we constructed the 95% bootstrap percentile equal-tailed pointwise and simultaneous confidence intervals for the overall mean µ and for the seasonal means µ₁, . . . , µ_d, re- spectively. We considered 2 sample sizesn= 240 andn= 480 and 5 different block lengths: b∈ {2,5,10,20,40} (for n= 240) b∈ {5,10,20,40,80}. The period lengthsd∈ {4,12,24}were chosen to represent periodicity often met in real data situations like hourly, monthly and quarterly. Number of boot- strap samples wasB = 500 and number of iterations was 500. Moreover, we took thew function of the form

w_c(t) =





t

c fort∈[0, c]

1 fort∈[c,1−c]

(1−t)

c fort∈[1−c,1]

withc= 0.43.Finally, the actual coverage probabilities (ACPs) of the 95%

equal tailed bootstrap pointwise confidence intervals for the overall mean and simultaneous confidence intervals for the seasonal means were calcu- lated. Results are presented in Figures 1-2.

In the overall mean estimation problem with n = 240 the ACPs are too low, which means that the confidence intervals obtained with the GSBB and the GSTBB are too narrow. For n= 480 the highest ACPs values are around 94% for the GSTBB, while for the GSBB they are 1-2% lower. It is worth to note that for most cases ACP curves for the GSTBB seem to be flatter than corresponding ones obtained with the GSBB. The highest dif- ference between the ACP values is observed forb= 80, d= 12 andn= 480 and is equal around 6%. Independently on the sample size and the chosen block length the GSTBB almost always outperforms the GSBB. It provides the ACPs that are closer to the nominal one.

Regarding simultaneous confidence intervals the ACP curves are quite flat independently on the period length and the sample size and those obtained with the GSTBB seem to be flatter. For example for n= 240 with d= 12 the ACPs obtained with the GSTBB differ from 95% less than 1%, while the GSBB ones range from 93% to 97%. In a few cases the differences between the performance of the GSBB and the GSTBB are small, but in general the GSTBB provides ACPs closer to 95%. The maximal difference is again observed forn= 480 withd= 12. For block lengthb= 40 it is equal around 3%. For the largest block length b = 80 independently on the sample size the ACPs obtained with the GSTBB are always higher and more accurate than the GSBB ones.

5 Appendix

Proof of Theorem 2.1. Under conditions of Theorem 2.1 Dudek et al.

(2014a) [3] showed consistency of GSBB for the overall mean and the sea- sonal means. In the sequel we follow the main idea of their proof, presenting only the main differences.

Without loss of generality we assume that the sample size n is an integer multiple of the block length b (n = lb) and is an integer multiple of the period length d (n = vd). We consider circular versions of the GSBB and the GSTBB.

At first we show (4). Let Z_t,b^∗ be the sum of observations contained in the

0 10 20 30 40 0.84

0.86 0.88 0.90 0.92 0.94 0.96 0.98

0 20 40 60 80

0.86 0.88 0.90 0.92 0.94 0.96

0 10 20 30 40

0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98

0 20 40 60 80

0.86 0.88 0.90 0.92 0.94 0.96

0 10 20 30 40

0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98

0 20 40 60 80

0.86 0.88 0.90 0.92 0.94 0.96

Figure 1: ACPs of pointwise equal-tailed percentile bootstrap confidence intervals for µvs. block length b. The three rows correspond to d= 4,12,24 respectively. Left and right column sample sizen = 240 andn= 480, respectively. GSBB method (grey) and GSBB-TBB (black). Nominal coverage probability is 95%.

0 10 20 30 40 0.84

0.86 0.88 0.90 0.92 0.94 0.96 0.98

0 20 40 60 80

0.86 0.88 0.90 0.92 0.94 0.96

0 10 20 30 40

0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98

0 20 40 60 80

0.86 0.88 0.90 0.92 0.94 0.96

0 10 20 30 40

0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98

0 20 40 60 80

0.86 0.88 0.90 0.92 0.94 0.96

Figure 2: ACPs of simultaneous equal-tailed percentile bootstrap confidence intervals for µi (i = 1, . . . , d) vs. block length b. The three rows correspond to d = 4,12,24 respectively. Left and right column sample sizen= 240 andn= 480, respectively. GSBB method (grey) and GSBB-TBB (black). Nominal coverage probability is 95%.

block of the lengthb, starting with observationY_t^∗, i.e Z_t^∗=Y_t^∗+· · ·+Y_t+b^∗ ₋₁

and Ze_t,b^∗ be a corresponding sum but obtained with the GSBB method i.e.

Ze_t^∗ =X_t^∗+· · ·+X_t+b−1^∗ . Note that E^∗Ze_t^∗ = 0 and E^∗Z_t^∗ = 0.

As in the proof of Theorem 1 in Dudek et al. (2014a) [3] we use Corollary 2.4.8 from Araujo and Gin´e (1980) [1]. Thus, we need to show that for any δ >0

l−1

∑

k=0

P^∗ ( 1

√nZ_1+kb,b^∗ > δ )

−→p 0, (7)

l−1

∑

k=0

E^∗ ( 1

√nZ_1+kb,b^∗ 1|^Z1+kb,b^∗ |^>√nδ ) p

−→0, (8)

l−1

∑

k=0

Var^∗ ( 1

√nZ_1+kb,b^∗ 1|^Z1+kb,b^∗ |^≤√nδ ) p

−→σ², (9) whereσ² is the asymptotic variance ofL(√

v(µb−µ)).

At first note that for eachk= 0, . . . , l−1 ands= 1, . . . , vE1/√

bZ1+kb+sd,b⁴ are uniformly bounded by constant independent on n, where

Z_1+kb+sd =Y_1+kb+sd+· · ·+Y_(k+1)b+sd and

Yj+kb+sd =wb(j)

√b

||w_b||2

Xemb+sd+j for j = 1, . . . , b.

This can be shown following the same reasoning as in the proofs of The- orems 1 and 3 from Kim (1994) [9]. Following main steps of the proof of Theorem 1 from Dudek et al. (2014a) [3] one can get (7) and (8). For (9) we additionally need to use Lemma 5 from Le´skow and Synowiecki (2010) [11]

for the arrayQn,s = ¹_bZ_s,b² , s∈S, whereS ={1 +kb+td: 1 +kb+td≤ n+b−1, t= 0, . . . , v−1, k= 1, . . . , l}. Note that this array isα-mixing with α_Q(τ) ≤ α_X(τ−b+ 1). Moreover, its elements have uniformly bounded second moments. Denote byn0 the number of elements of setS. Addition- ally, we define the array Qe_n,s= ¹_bZe_s,b² , s∈S, where

Ze_1+kb+sd =Xe_1+kb+sd+· · ·+Xe_(k+1)b+sd.

Dudek et al. (2014a) [3] showed that (1/n₀)∑

s∈SE (Qe_n,s

)→σ² (see proof of Theorem 1). We use this fact to obtain the same property for the array Qn,s. We have that

1 n0

∑

a∈S

E (Q_n,a) = 1 n0

∑

a∈S

Var ( 1

√bZ_a,b )

. Moreover, for anyk= 0, . . . , l−1 andt= 1, . . . , v

Var ( 1

√bZ_1+kb+td,b )

= 1

||w||²₂E ( _b

∑

i=1

w_b(i)Xe_i+kb+td )2

=

= 1

||w||²₂

∑b i=1

b−1

∑

j=−b+1

w_b(i)w_b(i+|j|)E

(Xe_i+kb+tdXe_i+|j|+kb+td )

=

= 1

||w||²₂

∑b i=1

b−1

∑

j=−b+1

w_b(i)w_b(i+|j|)E

(Xe_i+kbXe_i+|j|+kb )

=

= 1

||w||²₂

∑d s=1

v∑s−1 m=0

b−1

∑

j=−b+1

wb(s+md)wb(s+md+|j|)E

(Xes+md+kbXe_s+md+|j|+kb )

=

= 1

||w||²₂

∑d s=1

v∑s−1 m=0

b−1

∑

j=−b+1

w_b(s+md)w_b(s+md+|j|)E

(Xe_s+kbXe_s+|j|+kb )

, where fors= 1, . . . , d v_s is the number of elements of the set {a=s+md: a≤b, m= 0,1, . . .}.

Sincew∗w is twice continuously differentiable at 0, we have (forj << b)

v∑s−1 m=0

w_b(s+md)w_b(s+md+|j|)∼vs(w∗w) (j

b )

∼vs(w∗w) (0). (10) By symbol∼we denote asymptotic equivalence, i.e. sequencesa_1,n, a_2,n are asymptotically equivalent a_1,n∼a_2,n ifa_1,n/a_2,n→1 as n→ ∞.

Additionally,||w||²2∼b(w∗w)(0).Thus, b

||w||²₂

v∑s−1 m=0

w_b(s+md)w_b(s+md+|j|)∼vs

and Var

( 1

√bZ_1+kb+td,b )

−Var ( 1

√bZe_1+kb+td,b) ≤

≤ 1 b

∑d s=1

b−1

∑

j=−b+1

E

(Xe_s+kbXe_s+|j|+kb)

b

||w||²₂

v∑s−1 m=0

w_b(s+md)w_b(s+md+|j|)−vs

≤

≤Cv_s b

∑d s=1

b−1

∑

j=−b+1

α_X(|j|)

b vs||w||²2

v∑s−1 m=0

w_b(s+md)w_b(s+md+|j|)−1 =

=Cv_s b

∑d s=1

kb

∑

j=−kb

α_X(|j|)

b vs||w||²2

v∑s−1 m=0

w_b(s+md)w_b(s+md+|j|)−1 + +Cv_s

b

∑d s=1

−∑kb−1 j=−b+1

α_X(|j|)

b vs||w||²2

v∑s−1 m=0

w_b(s+md)w_b(s+md+|j|)−1 + +Cv_s

b

∑d s=1

b−1

∑

j=kb+1

αX(|j|)

b v_s||w||²₂

v∑s−1 m=0

wb(s+md)wb(s+md+|j|)−1 =

=I+II+III,

whereC is some positive constant independent on n, k_b/b →0 as n→ ∞. Note thatv_s/b→1/dasn→ ∞. To get the convergence to 0 ofI one needs to use (10) andα-mixing property of Xt. Using the fact that the absolute value in the second and the third summand is bounded and the time series isα-mixing, one gets convergence to 0 ofII andIII. Finally, we get that

sup

k,t

Var ( 1

√bZ1+kb+td,b

)

−Var ( 1

√b

Ze1+kb+td,b

)−→0,

and

1 n0

∑

a∈S

E (Q_n,a)→σ²,

which gives us the desired convergence in probability of 1/n₀∑

s∈SE (Q_n,s) to σ². The remaining steps of proof of (9) are the same as presented by Dudek et al. (2014a) [3] (see Theorem 1), so we omit the details.

Finally, to get (3) one needs to follow the proof of Theorem 1 from Dudek et al. (2014a) [3] applying the changes as in the above. Thus, again we omit

the details.

Proof of Theorems 2.2 and 2.3. Since the reasoning follows exactly the same steps as presented by Dudek et al. (2014a) [3] (see proofs of The- orems 4.2 and 4.3), we omit technical details.

Proof of Theorem 3.1. We give a sketch of the proof only for the real part ofa(λ, τ). For the imaginary part the reasoning follows the same steps.

Finally, the multidimensional consistency can be obtained from the Cram´er- Wold device. We need to show that

sup

x∈R

P(√

n(ℜ(ba_n(λ, τ))− ℜ(a(λ, τ)))≤x))

−

−P^∗(√

n(ℜ(ba^∗_n(λ, τ))−E^∗(ℜ(ba^∗_n(λ, τ))))≤x)−→^p 0.

Similarly to Dudek et al. (2014b) [4] we do not show consistency ofℜ(ba^∗_n(λ, τ)) directly, but we use asymptotically equivalent estimator of the form

e

a^∗_n(λ, τ) = 1 n

l−1

∑

k=0 b−τ

∑

m=1

Y˘_m+kb^∗ .

Note that ∑_b−τ

m=1ea^∗_n(λ, τ) is based only on elements contained in the k-th block, which is of the form (X_1+kb^∗ , . . . , X_b+kb^∗ ). Estimator ea^∗_n(λ, τ) was ob- tained form ba^∗_n(λ, τ) by removing those summands ˘Y_t^∗, for which X_t^∗ and X_t+τ^∗ belong to two consecutive blocks. To get asymptotic equivalence of ea^∗_n(λ, τ) andba^∗_n(λ, τ), we need to show that

√n|ℜ(ba^∗_n(λ, τ))− ℜ(ea^∗_n(λ, τ))−E^∗(ℜ(ba^∗_n(λ, τ)))−E^∗(ℜ(ea^∗_n(λ, τ)))|−→^p∗ 0.

By Tchebychev’s inequality it is enough to show the convergence of variance nVar^∗(ℜ(ba^∗_n(λ, τ))− ℜ(ea^∗_n(λ, τ)))−→^p∗ 0. (11) To get (11) one needs to follow the reasoning proposed in Dudek et al.

(2014b) [4] (see proof of (7.1)) and hence we omit the technical details.

Now it is enough to prove consistency ofℜ(ea^∗_n(λ, τ)), i.e.

x∈Rsup P(√

n(ℜ(ba_n(λ, τ))− ℜ(a(λ, τ)))≤x))

−

−P^∗(√

n(ℜ(ba^∗_n(λ, τ))−E^∗(ℜ(ba^∗_n(λ, τ))))≤x)−→^p 0.

To do that one needs to use the same arguments as in the proof of Theorem 2.1. The necessary facts like convergence of the variance can be found in

Dudek et al. (2014b) [4]. Since the whole reasoning follows exactly the same

steps, we again skip the details.

Acknowledgements

For this project, Anna Dudek has received funding from the Euro- pean Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 655394. Research of Dimitris Politis was partially sup-

ported by NSF grants DMS-12-23137 and DMS-13-08319.

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