Anna E Dudek
email: aedudek@agh.edu.pl

First and second order analysis for periodic random arrays using block bootstrap methods

Keywords: block bootstrap, consistency, Fourier coefficients of mean and autocovariance functions, periodic triangular array

First and second order analysis for periodic random arrays using block bootstrap methods

Anna E.

Introduction

In the recent years the number of bootstrap applications for periodic processes has constantly grown. An important class of periodic processes are periodically correlated (PC) processes, which were introduced by [START_REF] Gladyshev | Periodically Correlated Random Sequences[END_REF]. They are widely used for modeling in many different settings such as climatology, hydrology, mechanics, vibroacoustics and economics. Many motivating examples can be found in [START_REF] Gardner | Cyclostationarity: Half a century of research[END_REF] Hurd and Miamee (2007), [START_REF] Antoni | Cyclostationarity by examples[END_REF] and [START_REF] Napolitano | Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications[END_REF]. Time series X t is called PC with period d if it has periodic mean and auto-1 covariance functions, i.e. E (X t+d ) = E (X t ) and B(t, τ ) = Cov (X t , X t+τ ) = Cov (X t+d , X t+τ +d ) [START_REF] Antoni | Cyclostationarity by examples[END_REF] for each t, τ ∈ Z. The period d is taken as the smallest positive integer such that (1) holds. There are two block bootstrap methods that can be applied for PC processes. These are the Moving Block Bootstrap (MBB) and the Generalized Seasonal Block Bootstrap (GSBB). The first method was introduced independently by [START_REF] Künsch | The jackknife and the bootstrap for general stationary observations[END_REF] and [START_REF] Liu | Moving blocks jacknife and bootstrap capture weak dependence[END_REF]. It is very general and it does not preserve the periodic structure contained in the data. As a result the number of possible applications of this technique for PC time series is limited. On the other hand, the GSBB of Dudek et al. (2014a) was designed for periodic processes. It keeps the periodicity but to apply it one needs to know the period length. The GSBB is the generalization of two block bootstrap methods: the Seasonal Block Bootstrap (SBB) [START_REF] Politis | Resampling time series with seasonal components[END_REF]) and the Periodic Block Bootstrap [START_REF] Chan | Block bootstrap estimation of the distribution of cumulative outdoor degradation[END_REF]). The first and the second order characteristics of PC time series that are often considered can be split into two groups depending on the domain in which the analysis is performed. In the time domain one can be interested, for example, in the overall mean and the seasonal means. In the frequency domain these are Fourier coefficients of the mean and the autocovariance functions. For all mentioned characteristics consistency of the MBB and the GSBB has been already shown (see [START_REF] Synowiecki | Consistency and application of moving block bootstrap for nonstationary time series with periodic and almost periodic structure[END_REF], Dudek et al. (2014a), Dudek et al. (2014b), [START_REF] Dudek | Circular block bootstrap for coefficients of autocovariance function of almost periodically correlated time series[END_REF]). PC processes can be applied when the period length is constant. In this paper we focus on a different case, i.e. when the period length is changing over time. A very important example illustrating this phenomena is a chirp signal. It is a signal in which frequency increases or decreases over time. It can be described by the following equation

χ(u) = S(u) exp(iϕ(u)),
where S(u) is a positive, low-pass, smooth amplitude function whose evolution is slow when compared to the oscillations of the phase ϕ(u). Chirps are commonly met in nature. For example in audio signals (animal communication, echolocation), radar and sonar systems, astrophysics (gravitational waves radiated by coalescing binaries), mechanics and vibrations (e.g. car engines), medicine (EEG data -epileptic seizure) and seismography. For more details and examples we refer the reader to Flandrin (2001) and the references therein. In this paper we consider a case, when period length is growing in time. This corresponds to the so-called down-chirp signal. An example of such signal is presented in Figure 1. To model such phenomena periodic random arrays can be used. Bootstrap for triangular arrays with the period growing in time was first considered by [START_REF] Leśkow | On bootstrapping periodic random arrays with increasing period[END_REF] and later by Dudek et al. (2014a). In both papers only the problem of the estimation of the overall mean was discussed. Leśkow and Synowiecki showed the consistency of the Periodic Block Bootstrap and Dudek et al. of the Generalized Seasonal Block Bootstrap. In this paper we would like to extend the area of possible applications of bootstrap methods for the periodic random arrays to the Fourier coefficients of the mean and autocovariance functions. The paper is organized as follows. Section 2 contains the formulation of the problem. In Section 3 the algorithms of the circular version of the GSBB and the MBB are recalled. The bootstrap consistency results for the coefficients of the mean and the autocovariance functions for triangular arrays with growing period are presented in Section 4. In Section 5 the bootstrap-t confidence intervals are constructed and a simulated data example is presented. Finally, in Section 6 we discuss possible future extensions of our work. All proofs can be found in the Appendix.

Problem formulation

Let {X n,t : t = 1, . . . , m n } be an array of real valued random variables. We assume that it is row-wise periodically correlated (PC), i.e. in each row the mean function µ n (t) = E (X n,t ) and the autocovariance function

B n (t, τ ) = Cov (X n,t , X n,t+τ ) are periodic in variable t with period d n .
Below we present the Fourier decomposition of the mean and the autoco-variance functions in the n-th row of the triangular array. We have

µ n (t) = ∑ γn∈Γn b n (γ n ) exp(iγ n t), B n (t, τ ) = ∑ λn∈Λn,τ a n (λ n , τ ) exp(iλ n t),
where the sets

Γ n = {γ n : b n (γ n ) ̸ = 0}, Λ n,τ = {λ n : a n (λ n , τ ) ̸ = 0} are finite and Γ n , Λ n,τ ⊆ {2kπ/d n , k = 0, . . . , d n -1}.
Moreover, the coefficients b(γ) and a(λ, τ ) can be calculated using the following formulas

b n (γ n ) = 1 d n dn ∑ t=1 µ n (t) exp(-iγ n t), a n (λ n , τ ) = 1 d n dn ∑ t=1 B n (t, τ ) exp(-iλ n t). ( 2
)
and their estimators are of the form

b n (γ n ) = 1 m n mn ∑ t=1 X n,t exp(-iγ n t), (3)
a n (λ n , τ ) = 1 m n mn-max{τ,0} ∑ t=1-min{τ,0} (X n,t+τ -µ n (t + τ )) (X n,t -µ n (t)) exp(-iλ n t), ( 4
)
where µ n (t) = ∑ γn∈Γn b n (γ n ) exp(iγ n t). In the following we use the simplified version of (4). Without loss of generality we can assume that E(X n,t ) ≡ 0 and taking τ ≥ 0 we get

a n (λ n , τ ) = 1 m n mn-τ ∑ t=1 X n,t X n,t+τ exp(-iλ n t). ( 5
)
Detailed information concerning the coefficients of the mean and autocovariance functions and their estimators can be found in the papers of [START_REF] Hurd | Nonparametric time series analysis for periodically correlated processes[END_REF], (1991) and Hurd and Leśkow (1992a), (1992b).

Bootstrap algorithms

In this paper we apply two block bootstrap methods. However, these will not be the MBB and the GSBB in their standard form, but their circular versions. In this approach the data are treated as wrapped on the circle. This allows us to reduce the edge effects caused by the fact that in the MBB and the GSBB pseudo-samples the observations from the beginning and the end of the original sample appear less often than the other observations.

)
where k t is iid from a discrete uniform distribution

P (k t = t + vd) = 1 w for v = 0, 1, . . . , w -1.
Since we consider the circular version of GSBB, when t + vd > n we take the shifted observations t + vd -n. The second bootstrap method that we consider is the Circular Block Bootstrap of [START_REF] Politis | A circular block-resampling procedure for stationary data[END_REF], which is a circular version of the Moving Block Bootstrap. Comparing with the cGSBB the only difference is in the second step of the algorithm. We do not keep the periodic structure of the data, so we do not need to restrict the set of blocks which are selected.

CBB algorithm

1. Choose a (positive) integer block size b(< n). Then, sample size n can be split into l blocks of length b and a shorter block of length r i.e., n = lb + r, l ∈ N and r ∈ {0, . . . , b -1}.

For

t = 1, b + 1, 2b + 1, . . . , lb + 1, let (Y * t , Y * t+1 , . . . , Y * t+b-1 ) = (Y kt , Y kt+1 , . . . , Y kt+b-1 )
where k t is iid from a discrete uniform distribution

P (k t = j) = 1 n for j = 0, 1, . . . , n -1.
If j > n we take j -n instead.

Main results

In the first part of this section we use the cGSBB method to construct consistent estimators of the coefficients of the autocovariance function defined by formula [START_REF] Doukhan | Mixing. Properies and Examples[END_REF]. Then, the corresponding result for the coefficients of the mean function is presented. The second part is dedicated to the CBB method.

Let m n = l n b n + r n , where r n ∈ {0, . . . , b n -1} i.e., the observations in the n-th row of the considered triangular array {X n,t : t = 1, . . . , m n } can be split into l n blocks of the length b n and some remaining part of length r n . For the sake of simplicity we use notation r, l, b instead of r n , l n , b n , whenever it is possible. Moreover, without loss of generality we assume that m n = w n d n , w n ∈ N , i.e. each row of the considered array contains w n periods. To obtain our results additionally we need to assume the following conditions:

A1 let {X n,t : t = 1, . . . ,
∑ k=1 kα δ/(4+δ) n (k) = K < ∞; A5 sup n,k { k 1+ζ α 1/2 n (k) } < ∞ for some ζ > 0.
Condition A2 denotes that the period length d n and the number of observed periods w n are growing to infinity together with the sample size. Constant q in A4 will take value 1 or 2 depending on the considered case. α-mixing assumption A4 and A5 are technical and also appear in [START_REF] Synowiecki | Consistency and application of moving block bootstrap for nonstationary time series with periodic and almost periodic structure[END_REF]. These kind of conditions are also considered for usual PC time series (see e.g. [START_REF] Synowiecki | Consistency and application of moving block bootstrap for nonstationary time series with periodic and almost periodic structure[END_REF], Dudek et. al (2014a)). For more details on mixing we refer the reader to [START_REF] Doukhan | Mixing. Properies and Examples[END_REF].

As first we define three different bootstrap versions of a n (λ n , τ ).

a * n (λ n , τ ) = 1 m n mn-τ ∑ t=1 X * n,t X * n,t+τ exp(-iλ n t), ( 6
)
a * n (λ n , τ ) = 1 m n mn-τ ∑ t=1 X * n,t X * n,t+τ exp(-iλ n t * ), ( 7
)
a * n (λ, τ ) = 1 m n l-1 ∑ k=0 b-τ -1 ∑ j=0 X * n,1+kb+j X * n,1+kb+j+τ exp(-iλ n (1 + kb + j) * )(8) + 1 m n max{0,r-τ -1} ∑ j=0 X * n,1+kb+j X * n,1+kb+j+τ exp(-iλ n (1 + kb + j) * ),
where X * n,t for t = 1, . . . , m n is the cGSBB version of the n-th row of X n,t . Note that in the formulas ( 7) and ( 8) the symbol '*' is present additionally in the argument of the exponential function. In this case the time argument no longer varies from 1 to m n -τ , but corresponds to the time indices of observations chosen in the block selection process. To be more precise let us assume that the chosen block (X * n,t , . . . , X * n,t+b-1 ) is of the form (X n,kt , . . . , X n,kt+b-1 ). Then the values (t * , . . . , (t + b -1) * ) correspond to selected time indices, i.e. (t * , . . . ,

(t + b -1) * ) = (k t , . . . , k t + b -1).
Estimator [START_REF] Drake | Imputation of missing observations for heavy tailed cyclostationary time series[END_REF] is the most natural. It is constructed by replacing in formula [START_REF] Doukhan | Mixing. Properies and Examples[END_REF] the original sample by its bootstrap counterpart. As we show later this estimator is consistent, because the GSBB preserves the periodic structure in the bootstrap sample. This form of estimator was also applied in Dudek et al. (2014b) for the autocovariance function of PC time series. The second estimator uses ideas presented in [START_REF] Dudek | Circular block bootstrap for coefficients of autocovariance function of almost periodically correlated time series[END_REF]. In this paper results from Dudek et al. (2014b) are extended to the wider class of processes called almost periodically correlated. Since in this case the period usually does not exist, the GSBB cannot be used and hence Dudek considered the CBB. One may notice that ( 6) and ( 7) are equal for λ n ∈ Λ n,τ . Finally estimator [START_REF] Dudek | A generalized block bootstrap for seasonal time series[END_REF] was obtained by removing some summands from [START_REF] Dudek | Circular block bootstrap for coefficients of autocovariance function of almost periodically correlated time series[END_REF]. To be more precise [START_REF] Dudek | A generalized block bootstrap for seasonal time series[END_REF] has been got from [START_REF] Dudek | Circular block bootstrap for coefficients of autocovariance function of almost periodically correlated time series[END_REF] after substraction of those summands for which X * n,t and X * n,t+τ belong to two different blocks selected in the second step of the cGSBB algorithm. Let B * 1+kb for k = 0, . . . , l -1 be the block of length b of the form (X * n,1+kb , . . . , X * n,(k+1)b ). The last block of length r is denoted by B * 1+lb = (X * n,lb+1 , . . . , X * n,mn ). Since (X * n,1 , . . . , X * n,mn ) = (B * 1 , . . . , B * 1+lb ), we rewrite formula (7) using the new notation.

a * n (λ n , τ ) = 1 m n ∑ t̸ ∈C * b,τ X * n,t X * n,t+τ exp(-iλ n t * ) + 1 m n ∑ t∈C * b,τ X * n,t X * n,t+τ exp(-iλ n t * ) = a * n (λ, τ ) + 1 m n ∑ t∈C * b,τ X * n,t X * n,t+τ exp(-iλ n t * ) exp(-iλ n t * ),
where the set C * b,τ contains those time indices for which X * n,t and X * n,t+τ belong to different blocks

C * b,τ = { t : X * n,t ∈ B * 1+kb , X * n,t+τ ∈ B * 1+(k+1)b , k = 0, . . . , l -1 and 1 ≤ t, t + τ ≤ m n } .
Estimator ( 8) is the most convenient for practical applications among the proposed formulas. As we discuss in Section 5 construction of the bootstrap confidence intervals for a n (λ n , τ ) is computationally very expensive. In our study we will generate 500 000 bootstrap samples. Estimator [START_REF] Dudek | A generalized block bootstrap for seasonal time series[END_REF] allows us to reduce this cost substantially. To obtain ( 6) and ( 7) one needs to calculate the estimator value according to the appropriate formula for each generated bootstrap sample. Formula (8) allows for a different approach. Note that each summand in ( 8) is based on a different block B * 1+kb , k = 0, . . . , l. Thus, instead of using cGSBB to get (X * n,1 , . . . , X * n,mn ) and then calculating a * n (λ, τ ), one can create new random variables

Y b j = j+b-τ -1 ∑ t=j X n,t X n,t+τ exp(-iλ n t) and Y r j = j+r-τ -1 ∑ t=j X n,t X n,t+τ exp(-iλ n t)
for j = 1, . . . , m n . Y b j and Y r j represent these parts of estimator a n (λ n , τ ) that are based on blocks starting with observation X n,i of length b and r, respectively. As in cGSBB algorithm we treat data as wrapped on the circle, which means that if any time index t is bigger than m n we take t-m n instead. Now a * n (λ, τ ) can be rewritten as follows

a * n (λ, τ ) = 1 m n   l-1 ∑ j=0 Y b k 1+jb + Y r k 1+lb   ,
where k 1+kb , k = 0, . . . , l are iid random variables form step 2 of the cGSBB algorithm (see Section 3). Thus, when we calculate values of Y b j and Y r j , we can very easy obtain many values of a * n (λ, τ ). It is enough to generate the appropriate number of strings k 1+kb , k = 0, . . . , l and sum the corresponding Y b k 1+kb and Y r k 1+lb values, which is computationally very cheap.

Below we present the cGSBB consistency result. For the sake of simplicity we formulate theorem using notation a * n (λ n , τ ), which represents any of the estimators ( 6)- [START_REF] Dudek | A generalized block bootstrap for seasonal time series[END_REF]. That means that a * n (λ n , τ ) can be replaced by any of the estimators ( 6)- [START_REF] Dudek | A generalized block bootstrap for seasonal time series[END_REF].

Theorem 4.1 Let {X n,t : t = 1, . . . , m n } be a triangular array such that E(X n,t ) ≡ 0. Assume that conditions A1-A3, A4(2), A5 hold. If b n → ∞ as n → ∞ such that b n = o(m n ), then the cGSBB is consistent i.e. sup x∈R P ( √ m n σ R n (ℜ a n (λ n , τ ) -ℜa n (λ n , τ )) ≤ x ) - -P * ( √ m n σ * R n (ℜa * n (λ n , τ ) -E * (ℜa * n (λ n , τ ))) ≤ x ) p -→ 0, ( 9
)
sup x∈R P ( √ m n σ I n (ℑ a n (λ n , τ ) -ℑa n (λ n , τ )) ≤ x ) - -P * ( √ m n σ * I n (ℑa * n (λ n , τ ) -E * (ℑa * n (λ n , τ ))) ≤ x ) p -→ 0, ( 10
)
where

σ 2R n = 1 d n dn ∑ t=1 ∞ ∑ s=-∞ B X n (t, s) cos(λ n t) cos(λ n (t + s)), ( 11
)
σ 2I n = 1 d n dn ∑ t=1 ∞ ∑ s=-∞ B X n (t, s) sin(λ n t) sin(λ n (t + s)), ( 12
)
and σ 2R n , σ 2I n are assumed to be positive. Moreover, X n,t = X n,t X n,t+τ (t = 1, . . . , m n -τ ) and B X n (t, τ ) is the autocovariance function in the n-th row. Additionally, σ * R n , σ * I n are the bootstrap counterparts of σ R n , σ I n .
The analogous result can be also formulated in terms of the coefficients of the mean function. In this case we consider two bootstrap estimators of (3):

b * n (γ n ) = 1 m n mn ∑ t=1 X * n,t exp(-iγ n t), ( 13
) b * n (γ n ) = 1 m n mn ∑ t=1 X * n,t exp(-iγ n t * ). ( 14
)
One may note that the estimator of (3) corresponding to ( 8) is exactly equal to [START_REF] Gladyshev | Periodically Correlated Random Sequences[END_REF]. Moreover, if m n = l n b n the estimator ( 14) obtained using the CBB method is unbiased, i.e. E * (b *

n (γ n )) = b n (γ n ). If additionally m n = w n d n
, its cGSBB version is also unbiased. One should be aware that this property does not hold for any of the bootstrap estimators ( 6)- [START_REF] Dudek | A generalized block bootstrap for seasonal time series[END_REF]. Using the CBB or the cGSBB ensures that each observation appears in the same number of blocks, but this fact allows us to provide unbiased estimator only when τ = 0. In this case estimator [START_REF] Dudek | Circular block bootstrap for coefficients of autocovariance function of almost periodically correlated time series[END_REF] is unbiased under the same conditions as just discussed for estimator [START_REF] Gladyshev | Periodically Correlated Random Sequences[END_REF]. If τ > 0, then the bias is caused by the fact that joining selected blocks to create a bootstrap pseudo-sample we introduce a dependence structure that was not present in the original sample. To calculate X * t X * t+τ for X * t and X * t+τ belonging to two consecutive bootstrap blocks, we often use observations that were not τ time units apart in the original data. This effect cannot be removed by using circular versions of block bootstrap methods.

In the theorem below we present bootstrap consistency for the coefficients of the mean function. For simplicity we use notation b * n (γ n ) to avoid the formulation of the result for ( 13) and ( 14) separately. Theorem 4.2 Let {X n,t : t = 1, . . . , m n } be a row-wise periodically correlated array of real valued random variables with period d n . Assume that conditions A2, A3, A4(1), A5 hold.

If b n → ∞ as n → ∞ such that b n = o(m n ), then GSBB is consistent i.e. sup x∈R P ( √ m n σ R n ( ℜ b n (γ n ) -ℜb n (γ n ) ) ≤ x ) - -P * ( √ m n σ * R n (ℜb * n (γ n ) -E * (ℜb * n (γ n ))) ≤ x ) p -→ 0, (15)
sup x∈R P ( √ m n σ I n ( ℑ b n (γ n ) -ℑb n (γ n ) ) ≤ x ) - -P * ( √ m n σ * I n (ℑb * n (γ n ) -E * (ℑb * n (γ n ))) ≤ x ) p -→ 0, ( 16
)
where

σ 2R n = 1 d n dn ∑ t=1 ∞ ∑ s=-∞ B X n (t, s) cos(λ n t) cos(λ n (t + s)), ( 17
)
σ 2I n = 1 d n dn ∑ t=1 ∞ ∑ s=-∞ B X n (t, s) sin(λ n t) sin(λ n (t + s)), ( 18
)
and σ 2R n , σ 2I n are assumed to be positive. Moreover, B X n (t, τ ) is the autocovariance function in n-th row of the array X n,t . Moreover,

σ * R n , σ * I n are the bootstrap counterparts of σ R n , σ I n .
The advantage of the GSBB method is that it preserves the periodic structure contained in the data. However, to use it we need to know the period length. In some applications (for more details we refer the reader to Napolitano (2012)) period length is not obvious. So far no research was done to prove consistency of the GSBB in such a case. Moreover, a study of the impact of the period length estimation error on the results should be performed. The alternative approach in such a situation is to apply the block bootstrap method that does not require knowledge of the period length such as the CBB. Please note that since the bootstrap sample obtained with the CBB is usually not periodic in moments, the estimator (6) in the general case is not consistent. Theorem 4.3 If in Theorems 4.1 and 4.2, instead of the cGSBB, the CBB is used and estimator [START_REF] Dudek | Circular block bootstrap for coefficients of autocovariance function of almost periodically correlated time series[END_REF] or ( 8) is considered, the assertions continue to hold.

Remark In this paper we consider the circular versions of the block bootstrap methods. The main reason for that is to reduce the edge effects. In the cGSBB and the CBB each observation is present in the same number of blocks. When the GSBB or the MBB is used, the observations from the beginning and the end of the sample appear in less blocks than the others. That can cause an additional bias. However, if in the theorems presented in this section the cGSBB and the CBB are replaced by the GSBB and the MBB respectively, all assumptions will hold. The changes in proofs will be minor and hence we do not present technical details.

Confidence intervals

In this section we construct the bootstrap equal-tailed confidence intervals for the real and the imaginary parts of a n (λ n , τ ) using the bootstrap-t method (see [START_REF] Efron | An Introduction to the Bootstrap[END_REF]). So far in all applications for PC processes the percentile bootstrap confidence intervals were constructed (see e. 2015)), because the bootstrap was applied to avoid variance estimation. In our simulation we substitute the variance estimator by its bootstrap counterpart (see chapter 6 in Efron and Tibshirani (1993)). Moreover, to calculate bootstrap quantiles we need to compute a bootstrap estimate of standard error for each bootstrap sample. Thus, we have two nested levels of bootstrap sampling (see p. 162 in Efron and Tibshirani (1993)). To be more precise for a given row of our triangular array {X n,t : t = 1, . . . , m n } and chosen set of frequencies λ n and τ we generate B = 1000 bootstrap samples. Then, for each sample we get ℜ a * ,i n (λ n , τ ) and ℑ a * ,i n (λ n , τ ), i = 1, . . . , B.

To obtain the bootstrap estimate of the standard error for each of the 1000 bootstrap samples, we generate B 1 = 500 bootstrap samples. For each new bootstrap sample we calculate again the bootstrap estimates ℜ a * ,i,j n (λ n , τ ) and ℑ a * ,i,j n (λ n , τ ), where i = 1, . . . , B and j = 1, . . . , B 1 . Finally,

σ * R,i n =   B 1 ∑ j=1 ( ℜ a * ,i,j n (λ n , τ ) -ℜ a * ,i,• n (λ n , τ ) ) 2 / (B 1 -1)   1/2 , ( 19
)
σ * I,i n =   B 1 ∑ j=1 ( ℑ a * ,i,j n (λ n , τ ) -ℑ a * ,i,• n (λ n , τ ) ) 2 / (B 1 -1)   1/2 , ( 20
)
where i = 1, . . . , B, ℜ a * ,i,• n (λ n , τ ) = 1/B 1 ∑ B 1 j=1 ℜ a * ,i,j n (λ n , τ ) and ℑ a * ,i,• n (λ n , τ ) = 1/B 1 ∑ B 1 j=1 ℑ a * ,i,j n (λ n , τ
). The quantiles are obtained using studentized statistics of the form

T * R (i) = ℜ a * i n (λ n , τ )-ℜ a * i,• n (λ n , τ ) σ * R,i n and T * I (i) = ℑ a * i n (λ n , τ )-ℑ a * i,• n (λ n , τ ) σ * I,i n , where i = 1, . . . , B.
Finally, for each point (λ n , τ ) the confidence intervals for ℜa n (λ n , τ ) are of the form

( a n (λ n , τ ) -t * R 0.975 σ * R n , a n (λ n , τ ) -t * R 0.025 σ * R n ) ,
where t * R 0.025 , t * R 0.975 are the 2.5% and 97.5% quantiles, respectively. Moreover,

σ * R n = ( B ∑ i=1 ( ℜ a * ,i n (λ n , τ ) -ℜ a * ,• n (λ n , τ ) ) 2 / (B -1) ) 1/2 , ( 21
)
where

ℜ a * ,• n (λ n , τ ) = 1/B ∑ B i=1 ℜ a * ,i n (λ n , τ
). The confidence interval for and ℑa n (λ n , τ ) is constructed correspondingly.

Below we provide an application of our results to the Doppler effect. The Doppler effect is a change in the frequency of the signal caused by contractions/dilations of time. It appears when the source (called the transmitter) emitting signal and observer (called the receiver) exhibit relative motion. It can be heard e.g. when a vehicle sounding a siren is approaching a person. In our study we consider continuous time process of the form

χ(u) = S(u) cos ( 2π
( f 0 + γ 2 u ) u ) = S(u) cos ( 2πf 0 u + πγu 2 ) ,
where u ∈ R and γ is chirp rate, i.e. the instantaneous rate of change of the frequency of a waveform. The chirp χ(u) describes the received signal in the case of the transmitted signal S(u) cos (2πf 0 u) and the relative motion between transmitter and receiver, with constant relative radial acceleration. A radial acceleration exists when an object moves on a curve. For more details we refer the reader to Napolitano (2012) Secs. 2.2.6.1 and 7.8.1.1. If S(u) is wide-sense stationary, then S(u) cos (2πf 0 u) is cyclostationary and χ(u) is generalized almost-cyclostationary (GACS). In the engineering language χ(u) is said to have a time-varying carrier frequency f 0 + γ 2 u. Let us consider the uniform sampling of χ(u) with sampling period T s = 1/f s and let us assume that it is sampled in several time intervals labeled by n = 1, . . . , N. In a generic time interval (u n,0 , u n,0 + g n T s ) the time-varying carrier frequency of the chirp modulated process χ(u) ranges from

f 0 + γ 2 u 0,n to f 0 + γ 2 (u 0,n + g n T s ). Under the assumption that |γ| 2 g n T s ≪ f 0 + |γ| 2 u n,0 (22)
the carrier frequency can be considered constant and equal to f 0 + γ 2 u n,0 in the whole interval. In such a case χ(u) for u ∈ (u n,0 , u n,0 + g n T s ) can be modeled as cyclostationary with period

T n,0 = 1 2 ( f 0 + γ 2 u n,0
) .

If T n,0 /T s = q n is rational and q n = c n /d n with c n and d n relatively prime then the discrete-time sampled signal

X n,t = χ(u)| u=tTs for u ∈ (u n,0 , u n,0 + g n T s )
is cyclostationary with period d n (see Izzo and Napolitano (1996) and(1997)).

If T n,0 and T s are incommensurate, then X n,t is almost cyclostationary with cycle frequencies kT s /T n,0 modulo 1, where k is an integer number.

In addition, it should be noticed that in order to satisfy [START_REF] Künsch | The jackknife and the bootstrap for general stationary observations[END_REF], the sample size of the block g n cannot be too large. Consequently, g n cannot be increased without a bound. Therefore, a limit exists to the best accuracy that can be achieved with the proposed technique in the considered example.

For our study the following values of parameters were set: In this time interval period length d n = 3. Moreover, the shift parameter τ = 0 and frequencies λ n ∈ {0, 0.01, . . . , 3.14}. To construct bootstrapt pointwise confidence intervals for ℜa n (λ n , 0) and ℑa n (λ n , 0) we used the cGSBB and the CBB methods. Four block lengths b were considered {3, 6, 9, 18}. Finally, since for each frequency we generate 0.5 mln samples, we use only estimator [START_REF] Dudek | A generalized block bootstrap for seasonal time series[END_REF].

f 0 = -0.
Let us recall that the set of true frequencies Λ n,0 ⊆ {2kπ/3, k = 0, 1, 2} = {0, 2/3π, 4/3π}. For the sake of clarity in the figures we present results in Hz, i.e. for λ n /(2π). Then, the just mentioned set is of the form {0, 1/3, 2/3} = {0, 0.(3), 0.( 6)}. Note that none of the considered frequencies is exactly equal to 0.(3) or 0.( 6). Thus, we expect to detect frequencies that are close to these values. For all considered block lengths b results are very similar and hence we restrict the presentation only to b = 3 (see Figure 2). Moreover, Table 1 contains the detected frequencies, i.e. those for which appropriate confidence intervals do not contain 0. One may notice that for the cGSBB we detect the same frequencies independently of the value of b. What is the most important is that these are exactly frequencies that are closest to the true ones. The situation is similar for the CBB used for ℑa n (λ n , 0). Even if the number of detected frequencies is not constant (for b = 3 we have 5 frequencies and for b ∈ {6, 9, 18} 3 frequencies), pointed frequencies are again the closest to the true ones. Additionally, for b ∈ {6, 9, 18} these are exactly the same frequencies as were detected using cGSBB. The situation is slightly different in the case of ℜa n (λ n , 0). We have 5 significant frequencies for b ∈ {3, 9, 18} and 6 for b = 6. We always detect 0Hz, 0.34Hz and 0.66Hz. These are frequencies that were also detected using cGSBB for ℜa n (λ n , 0). But among other frequencies we have definitely false detections: λ n = 0.5Hz (for b = 6), λ n = 0.39Hz and λ n = 0.61Hz (for b = 18). It seems that the CBB is more sensitive to the block length choice. However, one should remember that the CBB does not require knowledge of the period length. In the case when d n needs to be estimated, the estimation error can affect the performance of the GSBB significantly. Since the CBB does not use d n , it can be the best choice in many practical applications. cGSBB CBB 0.00 0.01 0.00 0.01 0.34 0.33 0.32 0.33 b = 3 0.66 0.67 0.34 0.34 0.66 0.66 0.68 0.67 0.00 0.01 0.00 0.01 0.34 0.33 0.32 0.33 b = 6 0.66 0.67 0.34 0.67 0.50 0.66 0.68 0.00 0.01 0.00 0.01 0.34 0.33 0.32 0.33 b = 9 0.66 0.67 0.34 0.67 0.66 0.68 0.00 0.01 0.00 0.01 0.34 0.33 0.34 0.33 b = 18 0.66 0.67 0.39 0.67 0.61 0.66 Table 1: For each value of block length b the detected frequencies (Hz) using 95% bootstrap-t equal-tailed pointwise confidence intervals for ℜa n (λ n , 0) (column 2 and 4) and ℑa n (λ n , 0) (column 3 and 5) are presented. Columns 2-3 are results for cGSBB, columns 4-5 results for CBB. together with bootstrap-t pointwise equal-tailed confidence intervals (black). First and second row results are for cGSBB and CBB, respectively. Nominal coverage probability is 95%, block length b = 3. On the x-axis frequencies λn/(2π).

Conclusions and open questions

In this paper we consider time series with growing period. Under some moments and mixing assumptions we provide consistency results for two block bootstrap approaches, namely the Circular Block Bootstrap and the circular version of the Generalized Seasonal Block Bootstrap. The usual MBB and GSBB can also be applied for our problem. The consistent bootstrap-t equal-tailed confidence intervals for the coefficient of the mean and the autocovariance function are constructed. However, the considered model is very important and can be applied in many different fields e.g. for chirp signals, there is a further need to continue research in this topic. In Section 5 we considered the simple chirp signal with stationary component. However, in some applications this part of the signal is assumed to be heavy-tailed (see [START_REF] Sahmoudi | IF estimation of multicomponent chirp signal in impulsive α-stable noise environment using parametric and non-parametric approaches[END_REF]). Recently appeared some results for heavy tailed PC processes (Gajecka-Mirek (2014), Drake et al. ( 2015), Gajecka-Mirek (2015)) and hence our future research will be dedicated to the extension of our results to these kind of models.

Appendix

Proof of Theorem 4.1

Without loss of generality we assume that m n is an integer multiple of the block length b n (m n = l n b n ). To simplify notation we denote b n = b.

To show asymptotic normality of the considered estimator we use Lemma 4 from [START_REF] Leśkow | On bootstrapping periodic random arrays with increasing period[END_REF]. We show details of the proof only for the real part of the estimator a n (λ n , τ ). The reasoning for the imaginary part is the same. Denote by {W n,t : t = 1, . . . , m n -τ } the array with elements are of the form W n,t = X n,t X n,t+τ cos (λ n t). Since X n,t is row-wise PC and X t is WP(4) we have that the array X n,t = X n,t X n,t+τ is row-wise PC with period d n . Moreover, cos (λ n t) is a periodic function with period d n . Thus, the array W n,t is also row-wise PC with period d n . Moreover, we show that

Var

( 1 √ m n mn-τ ∑ t=1 X n,t X n,t+τ cos (λ n t) ) -σ 2R n -→ 0. (23)

Note that

Var

( 1 √ m n mn-τ ∑ t=1 X n,t X n,t+τ cos (λ n t) ) = = 1 m n mn-τ ∑ t=1 mn-τ ∑ s=1 Cov ( X n,t , X n,s
) cos (λ n t) cos (λ n s) .

Let m n -τ = pd n + r, where p ∈ N and r ∈ {0, . . . , d n -1}. The right-hand side of the last equality we can rewrite as

1 m n pdn ∑ t=1 pdn ∑ s=1 Cov ( X n,t , X n,s ) cos (λ n t) cos (λ n s) + + 1 m n pdn+r ∑ t=pdn+1 pdn+r ∑ s=1 Cov ( X n,t , X n,s ) cos (λ n t) cos (λ n s) + + 1 m n pdn+r ∑ t=1 pdn+r ∑ s=pdn+1 Cov ( X n,t , X n,s ) cos (λ n t) cos (λ n s) . ( 24
)
The array X n,t is row-wise α-mixing with α X n (k) = α n (k -τ ) for k > τ and α X n (k) = α n (0) otherwise. First we show that the third summand is O(1/m n ). For the second summand the reasoning is the same so this case will be omitted. Note that

1 m n pdn+r ∑ t=1 pdn+r ∑ s=pdn+1 Cov ( X n,t , X n,s ) cos (λ n t) cos (λ n s) = = 1 m n pdn+1-τ ∑ t=1 pdn+r ∑ s=pdn+1 Cov ( X n,t , X n,s ) cos (λ n t) cos (λ n s) + + 1 m n pdn+r ∑ t=pdn+2-τ pdn+r ∑ s=pdn+1 Cov ( X n,t , X n,s ) cos (λ n t) cos (λ n s) = = 1 m n pdn+1-τ ∑ t=1 pdn+r ∑ s=pdn+1 Cov ( X n,t , X n,s ) cos (λ n t) cos (λ n s) + O ( 1 m n ) = = 1 m n pdn+r ∑ s=pdn+1 s-1 ∑ s 1 =s-pdn-1+τ Cov ( X n,s-s 1 , X n,s ) cos (λ n t) cos (λ n s) + O ( 1 m n ) .
Using Lemma A.0.

) ≤ 8 sup s ( E (X n,s ) 4+2δ ) 1/(2+δ) α n (s 1 -τ ) δ/(2+δ) (25)
and

1 m n pdn+r ∑ s=pdn+1 s-1 ∑ s 1 =s-pdn-1+τ Cov ( X n,s-s 1 , X n,s ) cos (λ n t) cos (λ n s) ≤ ≤ Cr m n ∞ ∑ s 1 =τ α n (s 1 -τ ) δ/(2+δ) = O ( 1 m n ) ,
where C is a positive constant independent of n. Moreover the first summand in [START_REF] Leśkow | On bootstrapping periodic random arrays with increasing period[END_REF] can be written as

1 m n pdn ∑ t=1 pdn ∑ s=1 Cov ( X n,t , X n,s ) cos (λ n t) cos (λ n s) = = 1 m n dn ∑ t=1 pdn ∑ s=-(p-1)dn+1 ( p - ⌊ s d n ⌋ ) Cov ( X n,t , X n,s ) cos (λ n t) cos (λ n s) = = p m n dn ∑ t=1 pdn ∑ s=-(p-1)dn+1 Cov ( X n,t , X n,s ) cos (λ n t) cos (λ n s) - - 1 m n dn ∑ t=1 pdn ∑ s=-(p-1)dn+1 ⌊ s d n ⌋ Cov ( X n,t , X n,s
) cos (λ n t) cos (λ n s) .

The absolute value of the last term on the right-hand side can be bounded from above by

1 d n dn ∑ t=1 1 m n mn ∑ s=-mn sCov ( X n,t , X n,s ) = = 1 d n dn ∑ t=1 1 m n mn-t ∑ s 1 =-mn-t (s 1 + t)Cov ( X n,t , X n,s 1 +t ) = = 1 d n dn ∑ t=1 1 m n t-τ ∑ s 1 =-mn-t (s 1 + t)Cov ( X n,t , X n,s 1 +t ) + + 1 d n dn ∑ t=1 1 m n t+τ ∑ s 1 =t-τ +1 (s 1 + t)Cov ( X n,t , X n,s 1 +t ) + + 1 d n dn ∑ t=1 1 m n mn-t ∑ s 1 =t+τ +1 (s 1 + t)Cov ( X n,t , X n,s 1 +t
) .

The summands of the right-hand side we denote by I, II and III, respectively.

Using [START_REF] Liu | Moving blocks jacknife and bootstrap capture weak dependence[END_REF] and assumption (ii) we have that

III ≤ 1 d n dn ∑ t=1 1 m n mn-t ∑ s 1 =t+τ +1 2s 1 Cov ( X n,t , X n,s 1 +t ) ≤ ≤ C 1 1 d n dn ∑ t=1 1 m n mn-t ∑ s 1 =t+τ +1 s 1 α n (s 1 -τ ) δ/(2+δ) ≤ ≤ C 1 1 d n dn ∑ t=1 1 m n ∞ ∑ s 1 =1 s 1 α n (s 1 ) δ/(2+δ) = O ( 1 m n ) ,
where C 1 is a positive constant independent of n. Analogously one can show that I = O(1/m n ). Additionally, using assumption (ii) we get that

II ≤ C 2 1 d n dn ∑ t=1 1 m n t+τ ∑ s 1 =t-τ +1 |s 1 + t| ≤ C 2 1 d n dn ∑ t=1 1 m n t+τ ∑ s 1 =t-τ +1 (2t + τ ) ≤ ≤ C 2 1 d n dn ∑ t=1 1 m n t+τ ∑ s 1 =t-τ +1 (2d n + τ ) = = C 2 1 m n (2d n + τ ) (2τ -1) = O ( 1 w n ) ,
where C 2 is a positive constant independent of n.

To get [START_REF] Lahiri | Resampling Methods for Dependent Data[END_REF] we need to show

p m n dn ∑ t=1 pdn ∑ s=-(p-1)dn+1 Cov ( X n,t , X n,s ) cos (λ n t) cos (λ n s) -σ 2R n -→ 0.
Since |p/m n -1/d n | → 0 as n → ∞, the expression on the left-hand side can be rewritten equivalently as

1 d n dn ∑ t=1   pdn-t ∑ s 1 =-(p-1)dn+1-t Cov ( X n,t , X n,s 1 +t ) cos (λ n (s 1 + t)) - - ∞ ∑ s=-∞ Cov ( X n,t , X n,t+s ) cos(λ n (t + s)) ) cos (λ n t) ≤ ≤ 1 d n dn ∑ t=1 -(p-1)dn-t ∑ s=-∞ Cov ( X n,t , X n,t+s ) + + 1 d n dn ∑ t=1 ∞ ∑ s=pdn-t+1 Cov ( X n,t , X n,t+s
) .

Using (25) the right-hand side of the above inequality can be bounded by

1 d n dn ∑ t=1 -(p-1)dn-t ∑ s=-∞ α n (s -τ ) δ/(2+δ) + 1 d n dn ∑ t=1 ∞ ∑ s=pdn-t+1 α n (s -τ ) δ/(2+δ) .
Form the assumption (ii) and Toeplitz Lemma the last expression tends to 0 as n → ∞ and simultaneously we get [START_REF] Lahiri | Resampling Methods for Dependent Data[END_REF]. As a result the W n,t /σ R n fulfills the assumptions of Lemma 4 from Leśkow and Synowiecki (2010) and

1 √ m n σ R n mn-τ ∑ t=1 W n,t d -→ N (0, 1) .
To prove the bootstrap consistency we use techniques developed in [START_REF] Dudek | Circular block bootstrap for coefficients of autocovariance function of almost periodically correlated time series[END_REF] and Dudek et al. (2014b). We start with the estimator a * n (λ, τ ) and a * n (λ, τ ). We show only (9) because proof of ( 10) is analogous. The real part of ℜ a * n (λ n , τ ) can be rewritten as following

ℜ a * n (λ, τ ) = 1 n l-1 ∑ k=0 Z * 1+bk,b ,
where

Z * t,b = X * n,t X * n,t+τ cos(λt * ) + • • • + X * n,t+b-τ -1 X * n,t+b-1 cos(λ(t + b -τ -1) * ).
(λ n , τ )) -ℜ ( a * n (λ n , τ )) -E * (ℜ (a * n (λ n , τ ))) -E * (ℜ ( a * n (λ n , τ )))| p * -→ 0.
The proof follows the same steps as the one presented in

Z t,b = Z t,b -E Z t,b , where Z t,b = X n,t X n,t+τ cos(λt) + • • • + X n,t+b-τ -1 X n,t+b-1 cos(λ(t + b -τ -1)). The bootstrap version of Z t,b is of the form Z * t,b = Z * t,b -E * Z * t,b .
For t = 1, b + 1, . . . , (l -1)b + 1 the random variables Z * t,b are conditionally independent (given the sample (X 1 , . . . , X n ) with common distribution

P * ( Z * t,b = Z t+kd,b ) = 1 w n for k = 0, . . . , w n -1.
By Corollary 2.4.8 from Araujo and Giné (1980) to get condition (9) we need to show that for any ν > 0

ln-1 ∑ k=0 P * ( 1 √ m n Z * 1+kb,b > ν ) p -→ 0, ( 26
) ln-1 ∑ k=0 E * ( 1 √ m n Z * 1+kb,b 1 |Z * 1+kb,b |> √ mnν ) p -→ 0, ( 27
) ln-1 ∑ k=0 Var * ( 1 √ m n Z * 1+kb,b 1 |Z * 1+kb,b |≤ √ mnν ) -σ 2R n p -→ 0. ( 28
)
The proof of ( 26)-( 28 ) -→ 0.

Note that

Var

( ln-1 ∑ k=0 1 √ b Z 1+kb,b ) = ln-1 ∑ k 1 =0 ln-1 ∑ k 2 =0 Cov ( 1 √ b Z 1+k 1 b,b , 1 √ b Z 1+k 2 b,b ) = = ln-1 ∑ k=0 Var ( 1 √ b Z 1+kb,b ) + ln-1 ∑ k 1 =0 ∑ k 2 ̸ =k 1 Cov ( 1 √ b Z 1+k 1 b,b , 1 √ b Z 1+k 2 b,b
) .

Additionally, since {W n,t } is row-wise periodically correlated we get

ln-1 ∑ k=0 1 w n wn-1 ∑ t=0 1 l n Var ( 1 √ b Z 1+kb+td,b ) = ln-1 ∑ k=0 1 l n Var ( 1 √ b Z 1+kb,b
) .

Thus, by Lemma A.0.1 of [START_REF] Politis | Subsampling[END_REF] we have

R n ≤ 1 l n ln-1 ∑ k 1 =0 ∑ k 2 ̸ =k 1 Cov ( 1 √ b Z 1+k 1 b,b , 1 √ b Z 1+k 2 b,b ) ≤ ≤ 1 l n ln-1 ∑ k 1 =0 ∑ k 2 ̸ =k 1 ( sup k E 1 √ b Z 1+kb,b 4
) 1/2 α 1/2 Y (|k 2 -k 1 | b -b + 1) ≤ ≤ C 3 ln-1 ∑ τ =1 α 1/2 n ((τ -1) b + 1) ,
where C 3 is some positive constant independent of n.

Under assumption (iii) the sum on the right-hand side can be bounded from above by

C 4 ln-1 ∑ τ =1 1 ((τ -1) b + 1) 1+ζ ≤ C 4 + C 4 ln-1 ∑ τ =2 1 (τ -1) 1+ζ b 1+ζ = O ( 1 b 1+ζ ) ,
where C 4 is some positive constants independent of n. Note that V is tending to 0 in probability. The condition (ii) for α-mixing function is corresponding to one used by Dudek et al. (2014a). But since the period length d n → ∞ for some ξ > 1/2 we have

1 l n ln-1 ∑ k=0 E ( 1 w n wn-1 ∑ t=0 1 √ b Z 1+kb+tdn,b ) 2 ≤ C w n wn-1 ∑ k=1 1 k ξ d ξ n + O ( 1 w n ) = = o ( 1
d ξ n ) + O ( 1 w n ) ,
which is the main difference compared to the arguments provided by Dudek et al. (2014a). This ends proof of theorem for ℜa * n (λ n , τ ) and ℜ a * n (λ n , τ ). Finally, to get corresponding result for ℜ a * n (λ n , τ ) it is enough to follow the same reasoning as presented above with essential modifications of notation and using the fact that rows of the triangular array are W P (4).

Proof of Theorem 4.2

Since the proof is the simplified version the reasoning presented in the proof of Theorem 4.1, it is omitted.

Proof of Theorem 4.3

The proof follows exactly the same steps as the proof of Theorem 4.1 and hence it is omitted. The main difference is the number of blocks that can be selected while the bootstrap sample is constructed. For the cGSBB this number is equal to w n , while for the CBB it is equal to the length of the sample m n .

Figure 1 :

 1 Figure 1: Example of down-chirp signal.

  g Dudek et al. (2014a), Dudek (2015), Dudek et al. (2014b), Dehay and Dudek (

Figure 2 :

 2 Figure 2: Estimators (red) of ℜan (λn, 0) (left column) and ℑan (λn, 0) (right column)

2 .

 2 ) is similar to the one presented inDudek et al. (2014b). Note that E 1/ √ bZ 1+kb+sd,b from Synowiecki (2007)). Following the reasoning of Dudek et al. (2014a) we get that the absolute expected value of (26) and (27) can be bounded from above by The terms on the right-hand side we denote by IV and V , respectively. Following the reasoning of Dudek et. al (2014a) and Leśkow and Synowiecki (2010) we get that |I -σ 2R n | tends to 0 in probability as n → ∞. The key step is to show that

  To simplify notation in this section we assume that Y t is a PC process with the known period d. Assume that (Y 1 , . . . , Y n ) is the observed sample. Moreover, let the sample length n be an integer multiple of the period length

	(Y * t , Y * t+1 , . . . , Y * t+b-1 ) = (Y kt , Y kt+1 , . . . , Y kt+b-1

d (n = wd, w ∈ N ). As first we present the circular version of GSBB (cGSBB). cGSBB algorithm 1. Choose a (positive) integer block size b(< n). Then, sample size n can be split into l blocks of the length b and a shorter block of the length r i.e., n = lb + r, l ∈ N and r ∈ {0, . . . , b -1}. 2. For t = 1, b + 1, 2b + 1, . . . , lb + 1, let

  where ε t are iid random variables from the standard normal distribution. We took values of χ(u) for u = 995.8, 996.8, . . . , 1048.8 and hence m n = 54.

	γ = 0.2	√ 19. Moreover, S(t) is a MA(2) process of the form	00005,
		S(t) = 12 + 0.2ε t-1 + 0.3ε t-2 + ε t ,

  1. form Politis et al. (1999) we get

	(
	Cov	X n,s-s 1 , X n,s

  Dudek et al. (2014b), so we omit technical details. In the sequel, when it is not confusing, to simplify notation we omit sometimes subscript n. Now it is enough to show[START_REF] Dudek | Generalized Seasonal Block Bootstrap in frequency analysis of cyclostationary signals[END_REF] for ℜ a * n (λ n , τ ). Similarly to[START_REF] Dudek | Circular block bootstrap for coefficients of autocovariance function of almost periodically correlated time series[END_REF] andDudek et al. (2014b) we define

is bounded (see[START_REF] Kim | Moment bounds for non-stationary dependent sequences[END_REF] and Lemma A.5.

Acknowledgements

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 655394.