Remigijus Leipus

Anne Philippe

Vytaute Pilipauskaite

Donatas Surgailis

Vytautė Pilipauskaitė

November

Sample covariances of random-coefficient

Keywords: Autoregressive model, Panel data, Mixture distribution, Long memory, Sample covariance, Scaling transition, Poisson random measure, Asymptotic self-similarity. 2010 MSC: 60F05, 62M10

come

Introduction

Dynamic panels providing information on a large population of heterogeneous individuals such as households, firms, etc. observed at regular time periods, are often described by simple autoregressive models with random parameters near unity. One of the simplest models for individual evolution is the random-coefficient AR(1) (RCAR(1)) process

X(t) = aX(t -1) + ε(t), t ∈ Z, (1.1)
with standardized i.i.d. innovations {ε(t), t ∈ Z} and a random autoregressive coefficient a ∈ [0, 1) independent of {ε(t), t ∈ Z}. Granger [START_REF] Granger | Long memory relationship and the aggregation of dynamic models[END_REF] observed that in the case when the distribution of a is sufficiently dense near unity the stationary solution of RCAR(1) equation in (1.1) may have long memory in the sense that the sum of its lagged covariances diverges. To be more specific, assume that the random coefficient a ∈ [0, 1) has a density function of the following form

φ(x) = ψ(x)(1 -x) β-1 , x ∈ [0, 1), (1.2)
where β > 0 and ψ(x), x ∈ [0, 1) is a bounded function with lim x↑1 ψ(x) =: ψ(1) > 0. Then for β > 1 the covariance function of stationary solution of RCAR(1) equation in (1.1) with standardized finite variance innovations decays as t -(β-1) , viz.,

γ(t) := EX(0)X(t) = E a |t| 1 -a 2 ∼ (ψ(1)/2)Γ(β -1)t -(β-1) , t → ∞, (1.3)
implying t∈Z | Cov(X(0), X(t))| = ∞ for β ∈ [START_REF] Beran | From short to long memory: Aggregation and estimation[END_REF][START_REF] Bhansali | Convergence of quadratic forms with nonvanishing diagonal[END_REF]. The same long memory property applies to the contemporaneous aggregate of N independent individual evolutions {X i (t)}, i = 1, . . . , N of (1.1) and the limit Gaussian aggregated process arising when N → ∞. Various properties of the RCAR(1) and more general RCAR equations were studied in Gonçalves and Gouriéroux [START_REF] Gonçalves | Aggrégation de processus autoregressifs d'ordre 1[END_REF], Zaffaroni [START_REF] Zaffaroni | Contemporaneous aggregation of linear dynamic models in large economies[END_REF], Celov et al. [START_REF] Celov | Time series aggregation, disaggregation and long memory[END_REF], Oppenheim and Viano [START_REF] Oppenheim | Aggregation of random parameters Ornstein-Uhlenbeck or AR processes: some convergence results[END_REF], Puplinskaitė and Surgailis [START_REF] Puplinskaitė | Aggregation of random coefficient AR(1) process with infinite variance and idiosyncratic innovations[END_REF], Philippe et al. [START_REF] Philippe | Contemporaneous aggregation of triangular array of randomcoefficient AR(1) processes[END_REF] and other works, see Leipus et al. [START_REF] Leipus | Aggregation and long memory: recent developments[END_REF] for review.

Statistical inference in the RCAR(1) model was discussed in several works. Leipus et al. [START_REF] Leipus | Orthogonal series density estimation in a disaggregation scheme[END_REF], Celov et al. [START_REF] Celov | Asymptotic normality of the mixture density estimator in a disaggregation scheme[END_REF] discussed nonparametric estimation of the mixing density φ(x) using empirical covariances of the limit aggregated process. For panel RCAR(1) data, Robinson [START_REF] Robinson | Statistical inference for a random coefficient autoregressive model[END_REF] and Beran et al. [START_REF] Beran | From short to long memory: Aggregation and estimation[END_REF] discussed parametric estimation of the mixing density. In nonparametric context, Leipus et al. [START_REF] Leipus | Nonparametric estimation of the distribution of the autoregressive coefficient from panel random-coefficient AR(1) data[END_REF] studied estimation of the empirical d.f. of a from panel RCAR [START_REF] Beran | From short to long memory: Aggregation and estimation[END_REF] observations and derived its asymptotic properties as N, n → ∞, while [START_REF] Leipus | Estimating long memory in panel randomcoefficient AR(1) data[END_REF] discussed estimation of β in (1.2) and testing for long memory in the above panel model. For a N × n panel comprising N samples {X i (t), t = 1, . . . , n} of length n, i = 1, . . . , N of independent RCAR [START_REF] Beran | From short to long memory: Aggregation and estimation[END_REF] processes in (1.1) with mixing distribution in (1.2), Pilipauskaitė and Surgailis [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF] studied the asymptotic distribution of the sample mean

XN,n := 1 N n N i=1 n t=1 X i (t) (1.4)
as N, n → ∞, possibly at a different rate. [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF] showed that for 0 < β < 2 the limit distribution of this statistic depends on whether N/n β → ∞ or N/n β → 0 in which cases XN,n is asymptotically stable with stability parameter depending on β and taking values in the interval (0, 2]. See Table 2 below. As shown in [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF], under the 'intermediate' scaling N/n β → c ∈ (0, ∞) the limit distribution of XN,n is more complicated and is given by a stochastic integral with respect to a certain Poisson random measure. The present paper discusses asymptotic distribution of sample covariances (covariance estimates)

γ N,n (t, s) := 1 N n 1≤i,i+s≤N 1≤k,k+t≤n

(X i (k) -XN,n )(X i+s (k + t) -XN,n ), (t, s) ∈ Z 2 , (1.5)
computed from a similar RCAR(1) panel {X i (t), t = 1, . . . , n, i = 1, . . . , N } as in [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF], as N, n jointly increase, possibly at a different rate, and the lag (t, s) ∈ Z 2 is fixed, albeit arbitrary. Particularly, for (t, s) = (0, 0), (1.5) agrees with the sample variance:

γ N,n (0, 0) = 1 N n N i=1 n k=1 (X i (k) -XN,n ) 2 .
(1.6)

The true covariance function γ(t, s) := EX i (k)X i+s (k + t) of the RCAR(1) panel model with mixing density in (1.2) exists when β > 1 and is given by

γ(t, s) =    γ(t), s = 0, 0, s = 0, (1.7)
where γ(t) defined in (1.3). Note that γ(t) cannot be recovered from a single realization of the nonergodic RCAR [START_REF] Beran | From short to long memory: Aggregation and estimation[END_REF] process {X(t)} in (1.1). However, the covariance function in (1.7) can be consistently estimated from the RCAR(1) N × n panel when N, n → ∞, together with rates. The limit distribution of the sample covariance may exist even for 0 < β < 1 when the covariance itself is undefined. As it turns out, the limit distribution of γ N,n (t, s) depends on the mutual increase rate of N and n, and is also different for temporal, or iso-sectional lags (s = 0) and cross-sectional lags (s = 0). The distinctions between the cases s = 0 and s = 0 are due to the fact that, in the latter case, the statistic in (1.5) involves products X i (k)X i+s (k + t) of independent processes X i and X i+s , whereas in the former case, X i (k) and X i (k + t) are dependent r.v.s.

The main results of this paper are summarized in Table 1 below. Rigorous formulations are given in Sections 3 and 4. For better comparison, Table 2 presents the results of [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF] about the sample mean in (1.4) for the same panel model.

Mutual increase rate of N, n Parameter region Limit distribution Table 1: Limit distribution of sample covariances γ N,n (t, s) in (1.5) Mutual increase rate of N, n Parameter region Limit distribution

N/n β → ∞ 0 < β < 2, β = 1 asymmetric β-stable N/n β → 0 0 < β < 2, β = 1 asymmetric β-stable N/n β → c ∈ (0, ∞) 0 < β < 2,
N/n β → ∞ 1 < β < 2 Gaussian 0 < β < 1 symmetric (2β)-stable N/n β → 0 0 < β < 2 symmetric β-stable N/n β → c ∈ (0, ∞) 0 < β < 2 'intermediate Poisson' Arbitrary β > 2 Gaussian
Table 2: Limit distribution of the sample mean XN,n in (1.4)

Remark 1.1. (i) β-stable limits in Table 1 a) arising when N/n β → 0 and N/n β → ∞ have different scale parameters and hence the limit distribution of temporal sample covariances is different in the two cases.

(ii) 'Intermediate Poisson' limits in Tables 1-2 refer to infinitely divisible distributions defined through certain stochastic integrals w.r.t. Poisson random measure. A similar terminology was used in [START_REF] Pilipauskaitė | Anisotropic scaling of random grain model with application to network traffic[END_REF].

(iii) It follows from our results (see Theorem 4.1 below) that a scaling transition similar as in the case of the sample mean [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF] arises in the interval 0 < β < 2 for temporal sample covariances and product random fields X v (u)X v (u + t), (u, v) ∈ Z 2 involving temporal lags, with the critical rate N ∼ n β separating regimes with different limit distributions. For 'cross-sectional' product fields X v (u)X v+s (u + t), (u, v) ∈ Z 2 , s = 0 involving cross-sectional lags, a similar scaling transition occurs in the interval 0 < β < 3/2 with the critical rate N ∼ n 2β between different scaling regimes, see Theorem 3.1. The notion of scaling transition for longrange dependent random fields in Z 2 was discussed in Puplinskaitė and Surgailis [START_REF] Puplinskaitė | Scaling transition for long-range dependent Gaussian random fields[END_REF], [START_REF] Puplinskaitė | Aggregation of autoregressive random fields and anisotropic long-range dependence[END_REF], Pilipauskaitė and Surgailis [START_REF] Pilipauskaitė | Anisotropic scaling of random grain model with application to network traffic[END_REF], [START_REF] Pilipauskaitė | Scaling transition for nonlinear random fields with long-range dependence[END_REF].

(iv) The limit distributions of cross-sectional sample covariances in the missing intervals 0 < β < 1/2 and 0 < β < 3/4 of Table 1 b) are given in Corollary 3.1 below. They are more complicated and not included in Table 1 b) since the term N n( XN,n ) 2 due to the centering by the sample mean in (1.5) may play the dominating role.

(v) We expect that the asymptotic distribution of sample covariances in the RCAR(1) panel model with common innovations (see [START_REF] Pilipauskaitė | Joint aggregation of random-coefficient AR(1) processes with common innovations[END_REF]) can be analyzed in a similar fashion. Due to the differences between the two models (the common and the idiosyncratic innovation cases), the asymptotic behavior of sample covariances might be quite different in these two cases.

(vi) The results in Table 1 a) are obtained under the finite 4th moment conditions on the innovations, see Theorems 4.1 and 4.2 below. Although the last condition does not guarantee the existence of the 4th moment of the RCAR(1) process, it is crucial for the limit results, including the CLT in the case β > 2. Scaling transition for sample variances of long-range dependent Gaussian and linear random fields on Z 2 with finite 4th moment was established in Pilipauskaitė and Surgailis [START_REF] Pilipauskaitė | Scaling transition for nonlinear random fields with long-range dependence[END_REF]. On the other side, Surgailis [START_REF] Surgailis | Stable limits of sums of bounded functions of long memory moving averages with finite variance[END_REF], Horváth and Kokoszka [START_REF] Horváth | Sample autocovariances of long-memory time series[END_REF] obtained stable limits of sample variances and autocovariances for long memory moving averages with finite 2nd moment and infinite 4th moment. Finally, we mention the important works of Davis and Resnick [START_REF] Davis | Limit theory for the sample covariance and correlation functions of moving averages[END_REF] and Davis and Mikosch [START_REF] Davis | The sample autocorrelations of heavy-tailed processes with applications to ARCH[END_REF] on limit theory for sample covariance and correlation functions of moving averages and some nonlinear processes with infinite variance, respectively.

The rest of the paper is organized as follows. Section 2 presents some preliminary facts, including the definition and properties of the intermediate processes appearing in Table 1. Section 3 contains rigorous formulations and the proofs of the asymptotic results for cross-sectional sample covariances (1.5), s = 0 and the corresponding partial sums processes. Analogous results for temporal sample covariances and partial sums processes are presented in Section 4. Section 4 also contains some applications of these results to estimation of the autocovariance function γ(t) in (1.3) from panel data. Some auxiliary proofs are given in Appendix.

Preliminaries

This section contains some preliminary facts which will be used in the following sections.

2.1. Double stochastic integrals and quadratic forms. Let B i , i = 1, 2 be independent standard Brownian motions (BMs) on the real line. Let

I i (f ) := R f (s)dB i (s), I ij (g) := R 2 g(s 1 , s 2 )dB i (s 1 )dB j (s 2 ), i, j = 1, 2, (2.1)
denote Itô-Wiener stochastic integrals (single and double) w.r.t. B i , B j . The integrals in (2.1) are jointly defined for any (non-random

) integrands f ∈ L 2 (R), g ∈ L 2 (R 2 ); moreover, EI i (f ) = EI ij (g) = 0 and EI i (f )I i (f ) =    0, i = i , f, f , i = i , f, f ∈ L 2 (R), (2.2) EI i (f )I i j (g) = 0, ∀i, i , j , f ∈ L 2 (R), g ∈ L 2 (R 2 ), EI ij (g)I i j (g ) =          0, (i, j) / ∈ {(i , j ), (j , i )}, g, g , (i, j) ∈ {(i , j ), (j , i )}, i = j, 2 g, symg , i = i = j = j , g, g ∈ L 2 (R 2 ),
where

f, f = R f (s)f (s)ds ( f := f, f ), g, g = R 2 g(s 1 , s 2 )g (s 1 , s 2 )ds 1 ds 2 ( g := g, g ) denote scalar products (norms) in L 2 (R) and L 2 (R 2 )
, respectively, and sym denotes the symmetrization, see, e.g., ( [START_REF] Giraitis | Large Sample Inference for Long Memory Processes[END_REF], sec. 11.5, 14.3). Note that for g(s

1 , s 2 ) = f 1 (s 1 )f 2 (s 2 ), f i ∈ L 2 (R), i = 1, 2 we have I ii (g) = I i (f 1 )I i (f 2 ) - f 1 , f 2 , I 12 (g) = I 1 (f 1 )I 2 (f 2 ), in particular, I 12 (g) = d f 1 f 2 Z 1 Z 2 , where Z i ∼ N (0, 1), i = 1, 2 are independent standard normal r.v.s. Let {ε i (s), s ∈ Z}, i = 1, 2 be independent sequences of standardized i.i.d. r.v.s, Eε i (s) = 0, Eε i (s)ε i (s ) = 1 if (i, s) = (i , s ), Eε i (s)ε i (s ) = 0 if (i, s) = (i , s ), i, i = 1, 2, s, s ∈ Z. Consider the centered quadratic form Q ij (h) = s 1 ,s 2 ∈Z h(s 1 , s 2 )[ε i (s 1 )ε j (s 2 ) -Eε i (s 1 )ε j (s 2 )], i, j = 1, 2, (2.3)
where h ∈ L 2 (Z 2 ). For i = j we additionally assume Eε

4 i (0) < ∞. Then the sum in (2.3) converges in L 2 and var(Q ij (h)) ≤ (1 + Eε 4 i (0)δ ij ) s 1 ,s 2 ∈Z h 2 (s 1 , s 2 ), (2.4)
see ([7], (4.5.4)). With any h ∈ L 2 (Z 2 ) and any α 1 , α 2 > 0 we associate its extension to L 2 (R 2 ), namely,

h (α 1 ,α 2 ) (s 1 , s 2 ) := (α 1 α 2 ) 1/2 h( α 1 s 1 , α 2 s 2 ), (s 1 , s 2 ) ∈ R 2 , (2.5)
with h (α 1 ,α 2 ) 2 = s 1 ,s 2 ∈Z h 2 (s 1 , s 2 ). We shall use the following criterion for the convergence in distribution of quadratic forms in (2.3) towards double stochastic integrals (2.1).

Proposition 2.1. ( [START_REF] Giraitis | Large Sample Inference for Long Memory Processes[END_REF], Proposition 11.5.5) Let i, j = 1, 2 and

Q ij (h α 1 ,α 2 ), α 1 , α 2 > 0 be a family of quadratic forms as in (2.3) with coefficients h α 1 ,α 2 ∈ L 2 (Z 2 ). For i = j we additionally assume Eε 4 i (0) < ∞. Suppose for some g ∈ L 2 (R 2 ) we have that lim α 1 ,α 2 →∞ h (α 1 ,α 2 ) α 1 ,α 2 -g = 0.
(2.6)

Then Q ij (h α 1 ,α 2 ) → d I ij (g) (α 1 , α 2 → ∞)
, where I ij (g) is defined as in (2.1).

The

dµ β ≡ µ β (dx 1 , dx 2 , dB 1 , dB 2 ) := ψ(1) 2 (x 1 x 2 ) β-1 dx 1 dx 2 P B (dB 1 )P B (dB 2 ), (2.7)
where β > 0 is parameter and P B is the Wiener measure on C(R). Let d M β := dM β -dµ β be the centered Poisson random measure. We shall often use finiteness of the following integrals:

R 2 + min 1, 1 x 1 x 2 (x 1 +x 2 ) (x 1 x 2 ) β-1 dx 1 dx 2 < ∞, ∀ 0 < β < 3/2, (2.8) R 2 + min 1, 1 x 1 +x 2 (x 1 x 2 ) β-2 dx 1 dx 2 < ∞, ∀ 1 < β < 3/2, (2.9) see Appendix. Let Y i (u; x) = u -∞ e -x(u-s) dB i (s), u ∈ R, x > 0, (2.10)
be a family of stationary Ornstein-Uhlenbeck (O-U) processes subordinated to

B i = {B i (s), s ∈ R}, B i , i = 1, 2 being independent BMs. Let z(τ ; x 1 , x 2 ) := τ 0 2 i=1 Y i (u; x i )du, τ ≥ 0, (2.11)
be a family of integrated products of independent O-U processes indexed by x 1 , x 2 > 0. We use the representation of (2.11)

z(τ ; x 1 , x 2 ) = R 2 τ 0 2 i=1 e -x i (u-s i ) 1(u > s i )du dB 1 (s 1 )
Z β (τ ) := L 1 z(τ ; x 1 , x 2 )dM β + L c 1 z(τ ; x 1 , x 2 )d M β , (2.13)
where

L 1 := {(x 1 , x 2 , B 1 , B 2 ) ∈ (R + × C(R)) 2 : x 1 x 2 (x 1 + x 2 ) ≤ 1}, L c 1 := (R + × C(R)) 2 \ L 1 (2.14)
and µ β (L 1 ) < ∞. For 1/2 < β < 3/2 the two integrals in (2.13) can be combined in a single one:

Z β (τ ) = (R + ×C(R)) 2 z(τ ; x 1 , x 2 )d M β . (2.15)
These and other properties of Z β are stated in the following proposition whose proof is given in the Appendix. We also refer to [START_REF] Rajput | Spectral representations of infinitely divisible processes[END_REF] and [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF] for general properties of stochastic integrals w.r.t. Poisson random measure.

Proposition 2.2. (i) The process Z β in (2.13) is well-defined for any 0 < β < 3/2. It has stationary increments, infinitely divisible finite-dimensional distributions, and the joint ch.f. given by

E exp i m j=1 θ j Z β (τ j ) = exp (R + ×C(R)) 2 e i m j=1 θ j z(τ j ;x 1 ,x 2 ) -1 dµ β , (2.16)
where θ j ∈ R, τ j ≥ 0, j = 1, . . . , m, m ∈ N. Moreover, the distribution of Z β is symmetric:

{Z β (τ ), τ ≥ 0} = fdd {-Z β (τ ), τ ≥ 0}. (ii) E|Z β (τ )| p < ∞ for p < 2β and EZ β (τ ) = 0 for 1/2 < β < 3/2.
(iii) For 1/2 < β < 3/2, Z β can be defined as in (2.15). Moreover, if 1 < β < 3/2, then EZ 2 β (τ ) < ∞ and

EZ β (τ 1 )Z β (τ 2 ) = (σ 2 ∞ /2) τ 2(2-β) 1 + τ 2(2-β) 2 -|τ 2 -τ 1 | 2(2-β) , τ 1 , τ 2 ≥ 0, (2.17)
where

σ 2 ∞ := ψ(1) 2 Γ(β -1) 2 /(4(2 -β)(3 -2β)). (iv) For 1/2 < β < 3/2, the process Z β has a.s. continuous trajectories. (v) (Asymptotic self-similarity) As b → 0, b β-2 Z β (bτ ) → fdd σ ∞ B 2-β (τ ), if 1 < β < 3/2, (2.18) b -1 (log b -1 ) -1/(2β) Z β (bτ ) → fdd τ V 2β , if 0 < β < 1, (2.19)
where

{B 2-β (τ ), τ ≥ 0} is a fractional Brownian motion with E[B 2-β (τ )] 2 = τ 2(2-β) , τ ≥ 0, 2 -β ∈ (1/2, 1), σ 2
∞ is given in (2.17), and V 2β is a symmetric (2β)-stable r.v. with ch.f. Ee

iθV 2β = e -c∞|θ| 2β , θ ∈ R, c ∞ := ψ(1) 2 2 1-2β Γ(β + (1/2))Γ(1 -β)/ √ π. For any 0 < β < 3/2, as b → ∞, b -1/2 Z β (bτ ) → fdd A 1/2 B(τ ), (2.20)
where A > 0 is a (2β/3)-stable r.v. with Laplace transform Ee -θA = exp{-σ 0 θ 2β/3 }, θ ≥ 0,

σ 0 := ψ(1) 2 2 -2β/3
z * (τ ; x) := τ 0 Y 2 (u; x)du, τ ≥ 0, x > 0, (2.22)
be integrated squared O-U processes. Note Ez * (τ ; x) = τ EY 2 (0; x) = τ 0 -∞ e 2xs ds = τ /(2x). We will use the representation

z * (τ ; x) = R 2 τ 0 2 i=1 e -x(u-s i ) 1(u > s i )du dB(s 1 )dB(s 2 ) + τ /(2x) (2.23)
as the double Itô-Wiener integral. The 'iso-sectional' intermediate process Z * β is defined for β ∈ (0, 2), β = 1 as stochastic integral w.r.t. the above Poisson measure, viz.,

Z * β (τ ) := R + ×C(R) z * (τ ; x)    dM * β , 0 < β < 1, d M * β , 1 < β < 2, τ ≥ 0. ( 2
E exp i m j=1 θ j Z * β (τ j ) = exp R + ×C(R) e i m j=1 θ j z * (τ j ;x) -1 -i m j=1 θ j z * (τ j ; x)1(1 < β < 2) dµ * β , (2.25)
where θ j ∈ R, τ j ≥ 0, j = 1, . . . , m, m ∈ N.

(ii) E|Z * β (τ )| p < ∞ for any 0 < p < β < 2, β = 1 and EZ * β (τ ) = 0 for 1 < β < 2. (iii) For 1 < β < 2, the process Z *
β has a.s. continuous trajectories.

(iv) (Asymptotic self-similarity) For any

0 < β < 2, β = 1, b -1 Z * β (bτ ) → fdd    τ V * β as b → 0, τ V + β as b → ∞, (2.26)
where V + β , V * β are a completely asymmetric β-stable r.v.s with ch.f.s Ee iθV + β = exp{ψ(1)

∞ 0 (e iθ/(2x) -1 - i(θ/(2x))1(1 < β < 2))x β-1 dx}, Ee iθV * β = exp{ψ(1) ∞ 0 E(e iθZ 2 /(2x) -1 -i(θZ 2 /(2x))1(1 < β < 2))x β-1 dx}, θ ∈ R and Z ∼ N (0, 1).
2.4. Conditional long-run variance of products of RCAR(1) processes. We use some facts in Proposition 2.4, below, about conditional variance of the partial sums process of the product

Y ij (t) := X i (t)X j (t) of two RCAR(1) processes. Split Y ij (t) = Y + ij (t)+Y - ij (t), where Y + ij (t) = s 1 ∧s 2 ≥1 a t-s 1 i a t-s 2 j 1(t ≥ s 1 ∨ s 2 )ε i (s 1 )ε j (s 2 ), Y - ij (t) = s 1 ∧s 2 ≤0 a t-s 1 i a t-s 2 j 1(t ≥ s 1 ∨ s 2 )ε i (s 1 )ε j (s 2 )
. For i = j we assume additionally that Eε 4 i (0) < ∞.

Proposition 2.4. We have

var n t=1 Y ij (t)|a i , a j ∼ var n t=1 Y + ij (t)|a i , a j ∼ A ij n, n → ∞, (2.27)
where

A ij :=    1+a i a j (1-a 2 i )(1-a 2 j )(1-a i a j ) , i = j, 1+a 2 i 1-a 2 i ( 2 (1-a 2 i ) 2 + cum 4 1-a 4 i ), i = j (2.28)
with cum 4 being the 4th cumulant of ε i (0). Moreover, for any n ≥ 1, i, j ∈ Z, a i , a j ∈ [0, 1)

var n t=1 Y ij (t)|a i , a j ≤ C ij n 2 (1 -a i )(1 -a j ) min 1, 1 n(2 -a i -a j ) , (2.29)
where

C ij := 4 (i = j), := 2(2 + |cum 4 |) (i = j). Proof. Let i = j. We have E[Y ij (t)Y ij (s)|a i , a j ] = E[X i (t)X i (s)|a i ]E[X j (t)X j (s)|a j ] = (a i a j ) |t-s| /(1 -a 2 i )(1 - a 2 j ) and hence J n (a i , a j ) := E n t=1 Y ij (t) 2 |a i , a j = n (1 -a 2 i )(1 -a 2 j ) n t=-n (a i a j ) |t| 1 - |t| n . (2.30) Relation (2.30) implies (2.27). It also implies J n (a i , a j ) ≤ 2n 2 /((1 -a i )(1 -a j )). Note also 1 -a i a j ≥ (1/2)((1 -a i ) + (1 -a j )).
Hence and from (2.30) we obtain

J n (a i , a j ) ≤ n (1 -a 2 i )(1 -a 2 j ) 1 + 2 ∞ t=1 (a i a j ) t ≤ 2n (1 -a i )(1 -a j )(1 -a i a j ) ≤ 4n (1 -a i )(1 -a j )(2 -a i -a j ) , proving (2.29). The proof of (2.27)-(2.29) for i = j is similar using cov[Y ii (t), Y ii (s)|a i ] = 2(a |t-s| i /(1 -a 2 i )) 2 + cum 4 a 2|t-s| i /(1 -a 4 i ).
3 Asymptotic distribution of cross-sectional sample covariances Theorems 3.1 and 3.2 discuss the asymptotic distribution of partial sums process

S t,s N,n (τ ) := N i=1 nτ u=1 X i (u)X i+s (u + t), τ ≥ 0, (3.1)
where t and s ∈ Z, s = 0 are fixed and N and n tend to infinity, possibly at a different rate. The asymptotic behavior of sample covariances γ N,n (t, s) is discussed in Corollary 3.1. As it turns out, these limit distributions do not depend on t, s which is due to the fact that the sectional processes {X i (t), t ∈ Z}, i ∈ Z are independent and stationary.

Theorem 3.1. Let the mixing distribution satisfy condition

(1.2) with 0 < β < 3/2. Let N, n → ∞ so as λ N,n := N 1/(2β) n → λ ∞ ∈ [0, ∞]. (3.2)
Then the following statements (i)-(iii) hold for S t,s N,n (τ ), (t, s) ∈ Z 2 , s = 0 in (3.1) depending on λ ∞ in (3.2). (i) Let λ ∞ = ∞. Then n -2 λ -β N,n S t,s N,n (τ ) → fdd σ ∞ B 2-β (τ ), 1 < β < 3/2, (3.3)
n -2 λ -1 N,n (log λ N,n ) -1/(2β) S t,s N,n (τ ) → fdd τ V 2β , 0 < β < 1, (3.4)
where the limit processes are the same as in (2.18), (

(ii) Let λ ∞ = 0 and E|ε(0)| 2p < ∞ for some p > 1.

Then

n -2 λ -3/2 N,n S t,s N,n (τ ) → fdd A 1/2 B(τ ), (3.5)
where the limit process is the same as in (2.20).

(iii) Let 0 < λ ∞ < ∞. Then n -2 λ -3/2 N,n S t,s N,n (τ ) → fdd λ 1/2 ∞ Z β (τ /λ ∞ ), (3.6)
where Z β is the intermediate process in (2.13).

Theorem 3.2. Let the mixing distribution satisfy condition (1.2) with β > 3/2 and assume E|ε(0)| 2p < ∞ for some p > 1. Then for any (t, s) ∈ Z 2 , s = 0 as N, n → ∞ in arbitrary way,

n -1/2 N -1/2 S t,s N,n (τ ) → fdd σB(τ ), σ 2 := EA 12 , (3.7)
where A 12 is defined in (2.28).

Remark 3.1. Our proof of Theorem 3.1 (ii) requires establishing the asymptotic normality of a bilinear form in i.i.d. r.v.s, which has a non-zero diagonal, see the r.h.s. of (3.52). For this purpose, we use the martingale CLT and impose an additional condition of E|ε(0)| 2p < ∞, p > 1. To establish the CLT for quadratic forms with non-zero diagonal, [START_REF] Bhansali | Convergence of quadratic forms with nonvanishing diagonal[END_REF] took similar approach and also needed 2p finite moments. In Theorem 3.2 we also assume E|ε(0)| 2p < ∞, p > 1. However, it can be proved under Eε 2 (0) < ∞ applying another technique that is approximation by m-dependent r.v.s. Moreover, this result holds if (1.2) is replaced by EA 12 < ∞.

Note that the asymptotic distribution of sample covariances γ N,n (t, s) in (1.5) coincides with that of the statistics γ N,n (t, s)

:= (N n) -1 S t,s N,n (1) -( XN,n ) 2 . (3.8)
For s = 0 the limit behavior of the first term on the r.h.s. of (3.8) can be obtained from Theorems 3.1 and 3.2.

It turns out that for some values of β, the second term on the r.h.s. can play the dominating role. The limit behavior of XN,n was identified in [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF] and is given in the following proposition, with some simplifications.

Proposition 3.1. Let the mixing distribution satisfy condition (1.2) with β > 0.

(i) Let 1 < β < 2 and N/n β → ∞. Then N 1/2 n (β-1)/2 XN,n → d σβ Z, (3.9)
where Z ∼ N (0, 1) and σ2

β := ψ(1)Γ(β -1)/((3 -β)(2 -β)). (ii) Let 0 < β < 1 and N/n β → ∞. Then N 1-1/2β XN,n → d V2β , (3.10)
where V2β is a symmetric (2β)-stable r.v. with ch.f.

Ee iθ V2β = e -Kβ |θ| 2β , Kβ := ψ(1)4 -β Γ(1 -β)/β. (iii) Let 0 < β < 2 and N/n β → 0. Then N 1-1/β n 1/2 XN,n → d Wβ , (3.11)
where Wβ is a symmetric β-stable r.v. with ch.f.

Ee iθ Wβ = e -kβ |θ| β , kβ := ψ(1)2 -β/2 Γ(1 -β/2)/β. (iv) Let β > 2.
Then as N, n → ∞ in arbitrary way,

N 1/2 n 1/2 XN,n → d σZ, (3.12)
where Z ∼ N (0, 1) and σ2 := E(1 -a) -2 .

From Theorems 3.1 and Proposition 3.1 we see that the r.h.s. of (3.8) may exhibit two 'bifurcation points' of the limit behavior, viz., as N ∼ n 2β and N ∼ n β . Depending on the value of β the first or the second term may dominate, and the limit behavior of γ N,n (t, s) gets more complicated. The following corollary provides this limit without detailing the 'intermediate' situations and also with exception of some particular values of β where both terms on the r.h.s. may contribute to the limit. Essentially, the corollary follows by comparing the normalizations in Theorems 3.1 and Proposition 3.1.

Corollary 3.1. Assume that the mixing distribution satisfies condition (1.2) with β > 0 and E|ε(0)| 2p < ∞ for some p > 1 and (t, s) ∈ Z 2 , s = 0 be fixed albeit arbitrary.

(i) Let N/n 2β → ∞ and 1 < β < 3/2. Then N 1/2 n β-1 γ N,n (t, s) → d σ ∞ Z,
where Z ∼ N (0, 1) and σ ∞ is the same as in Theorem 3.1 (i).

(ii) Let N/n 2β → ∞ and 1/2 < β < 1. Then N 1-1/(2β) log 1/(2β) (N 1/(2β) /n) γ N,n (t, s) → d V 2β , where V 2β is symmetric (2β)-stable r.v. defined in Theorem 3.1 (i). (iii) Let N/n 2β → ∞ and 0 < β < 1/2. Then N 2-1/β γ N,n (t, s) → d -( V2β ) 2 , (3.13)
where V2β is symmetric (2β)-stable r.v. defined in Proposition 3.1 (ii).

(iv) Let N/n 2β → 0, N/n β → ∞ and 3/4 < β < 3/2. Then

N 1-3/(4β) n 1/2 γ N,n (t, s) → d W 4β/3 , (3.14)
where W 4β/3 is a symmetric (4β/3)-stable r.v. with characteristic function Ee iθW 4β/3 = e -(σ 0 /2 2β/3 )|θ| 4β/3 and σ 0 is the same constant as in Theorem 3.1 (ii).

(v) Let N/n 2β → 0, 1/2 < β < 3/4 and N/n 2β/(4β-1) → ∞. Then the convergence in (3.14) holds.

(vi) Let N/n β → ∞, 1/2 < β < 3/4 and N/n 2β/(4β-1) → 0. Then the convergence in (3.13) holds.

(vii) Let N/n 2β → 0, N/n β → ∞ and 0 < β < 1/2. Then the convergence in (3.13) holds.

(viii) Let N/n β → 0 and 3/4 < β < 3/2. Then the convergence in (3.14) holds.

(ix) Let N/n β → 0, 0 < β < 3/4 and N/n 2β/(5-4β) → ∞. Then N 2-2/β γ N,n (t, s) → d -( Wβ ) 2 , (3.15)
where Wβ is a symmetric β-stable r.v. defined in Proposition 3.1 (iii).

(x) Let 0 < β < 3/4 and N/n 2β/(5-4β) → 0. Then the convergence in (3.14) holds.

(xi) For 3/2 < β < 2, let N/n β → [0, ∞] and for β > 2, let N, n → ∞ in arbitrary way. Then

N 1/2 n 1/2 γ N,n (t, s) → d N (0, σ 2 ), (3.16)
where σ 2 is given as in Theorem 3.2.

The proof of Theorem 3.1 in cases (i)-(iii) is given subsections 3.1-3.3. To avoid excessive notation, the discussion is limited to the case (t, s) = (0, 1) or the partial sums process S N,n (τ

) := N i=1 nτ t=1 X i (t)X i+1 (t).
Later on we shall extend them to general case (t, s), s = 0.

Let us give an outline of the proof of Theorem 3.1. Similarly to [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF] we use the method of characteristic function combined with 'vertical' Bernstein's blocks, due to the fact that S N,n is not a sum of row-independent summands as in [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF]. Write S N,n (τ ) = S N,n;q (τ ) + S † N,n;q (τ ) + S ‡ N,n;q (τ ), (3.17) where the main term

S N,n;q (τ ) := Ñq k=1 Y k,n;q (τ ), Y k,n;q (τ ) := (k-1)q<i<kq nτ t=1 X i (t)X i+1 (t), 1 ≤ k ≤ Ñq := N q , (3.18)
is a sum of Ñq 'large' blocks of size q -1 with

q ≡ q N,n → ∞ as N, n → ∞. (3.19)
The convergence rate of q ∈ N in (3.19) will be slow enough (e.g., q = O(log N )) and specified later on. The two other terms in the decomposition (3.17), (3.20) contain respectively Ñq = o(N ) and N -q Ñq < q = o(N ) row sums and will be shown to be negligible. More precisely, we show that in each case (i)-(iii) of Theorem 3.1,

S † N,n;q (τ ) := Ñq k=1 nτ t=1 X kq (t)X kq+1 (t), S ‡ N,n;q (τ ) := q Ñq<i≤N nτ t=1 X i (t)X i+1 (t),
A -1 N,n S N,n;q (τ ) → fdd S β (τ ), (3.21)
A -1 N,n S † N,n;q (τ ) = o p (1), A -1 N,n S ‡ N,n;q (τ ) = o p (1), (3.22)
where A N,n and S β denote the normalization and the limit process, respectively, particularly,

A N,n := n 2          λ β N,n , λ ∞ = ∞, 1 < β < 3/2, λ N,n (log λ N,n ) 1/(2β) , λ ∞ = ∞, 0 < β < 1, λ 3/2 N,n , λ ∞ ∈ [0, ∞), 0 < β < 3/2. (3.23)
Note that the summands Y k,n;q , 1 ≤ k ≤ Ñq in (3.18) are independent and identically distributed, and the limit S β (τ ) is infinitely divisible in cases (i)-(iii) of Theorem 3.1. Hence use of characteristic functions to prove (3.21) is natural. The proofs are limited to one-dimensional convergence at a given τ > 0 since the convergence of general finite-dimensional distributions follows in a similar way. Accordingly, the proof of (3.21) for fixed τ > 0 reduces to

Φ N,n;q (θ) → Φ(θ), as N, n → ∞, λ N,n → λ ∞ , ∀θ ∈ R, (3.24)
where

Φ N,n;q (θ) := Ñq E e iθA -1 N,n Y 1,n;q (τ ) -1 , Φ(θ) := log Ee iθS β (τ ) . (3.25)
To prove (3.24) write

A -1 N,n Y 1,n;q (τ ) = q-1 i=1 y i (τ ), where y i (τ ) := A -1 N,n nτ t=1 X i (t)X i+1 (t). (3.26)
We use the identity:

1≤i<q (1 + w i ) -1 = 1≤i<q w i + |D|≥2 i∈D w i , (3.27)
where the sum |D|≥2 is taken over all subsets D ⊂ {1, . . . , q -1} of cardinality |D| ≥ 2. Applying (3.27) with w i = e iθy i (τ ) -1 we obtain Φ N,n;q (θ) := Ñq (q -1) Ee iθy 1 (τ ) -1 + Ñq |D|≥2 E i∈D e iθy i (τ ) -1 .

(3.28) Thus, since Ñq (q -1)/N → 1, (3.24) follows from

N Ee iθy 1 (τ ) -1 → Φ(θ), (3.29)
N |D|≥2 E i∈D e iθy i (τ ) -1 → 0. (3.30)
Let us explain the main idea of the proof of (3.29). Assuming φ(x) = (1 -x) β-1 in (1.2) the l.h.s. of (3.29) can be written as

N Ee iθy 1 (τ ) -1 = N (0,1] 2 E e iθy 1 (τ ) -1 a i = 1 -z i , i = 1, 2 (z 1 z 2 ) β-1 dz 1 dz 2 = N B 2β N,n (0,B N,n ] 2 E e iθz N,n (τ ;x 1 ,x 2 ) -1 (x 1 x 2 ) β-1 dx 1 dx 2 , (3.31)
where

z N,n (τ ; x 1 , x 2 ) := A -1 N,n s 1 ,s 2 ∈Z ε 1 (s 1 )ε 2 (s 2 ) nτ t=1 2 i=1 1 - x i B N,n t-s i 1(t ≥ s i ) (3.32)
and B N,n → ∞ is a scaling factor of the autoregressive coefficient. In cases (ii) and (iii) of Theorem 3.1 (proof of (3.5) and (3.6)) we choose this scaling factor

B N,n = N 1/(2β) so that N B 2β N,n
= 1 and prove that the integral in (3.31) converges to

R 2 + E e iθz(τ ;x 1 ,x 2 ) -1 (x 1 x 2 ) β-1 dx 1 dx 2 = Φ(θ)
, where z(τ ; x 1 , x 2 ) is a random process and Φ(θ) is the required limit in (3.24). A similar scaling

B N,n = (N log λ N,n ) 1/(2β) applies in the case λ ∞ = ∞, 0 < β < 1 (proof of (3. 4
)) although in this case the factor N/B 2β N,n = 1/ log λ N,n in front of the integral in (3.31) does not trivialize and the proof of the limit in (3.24) is more delicate. On the other hand, in the case of the Gaussian limit (3.3), the choice (2.11) as shown in subsection 3.3 below.

B N,n = n leads to N/B 2β N,n = λ 2β N,n → ∞ and (3.31) tends to (1/2)|θ| 2 R 2 + Ez 2 (τ ; x 1 , x 2 )(x 1 x 2 ) β-1 dx 1 dx 2 = Φ(θ) with z(τ ; x 1 , x 2 ) defined in
To summarize the above discussion: in each case (i)-(iii) of Theorem 3.1, to prove the limit (3.21) of the main term, it suffices to verify relations (3.29) and (3.30). The proof of the first relation in (3.22) is very similar to (3.21) since S † N,n;q (τ ) is also a sum of i.i.d. r.v.s and the argument of (3.21) applies with small changes. The proof of the second relation in (3.22) seems even simpler. In the proofs we repeatedly use the following inequalities:

|e iz -1| ≤ 2 ∧ |z|, |e iz -1 -iz| ≤ (2|z|) ∧ (z 2 /2), z ∈ R. (3.33) 3.1 Proof of Theorem 3.1 (iii): case 0 < λ ∞ < ∞
Proof of (3.29). For notational brevity, we assume λ N,n = λ ∞ = 1 since the general case as in (3.2) requires unsubstantial changes. Recall from (2.16) that Φ(θ) =

R 2 + E[e iθz(τ ;x 1 ,x 2 ) -1](x 1 x 2 ) β-1 dx 1 dx 2
, where z(τ ; x 1 , x 2 ) is the double Itô-Wiener integral in (2.11). Also recall the representation (3.31), (3.32), where

A N,n = n 2 , B N,n = n and z N,n (τ ; x 1 , x 2 ) = Q 12 (h n (•; τ ; x 1 , x 2 )) is a quadratic form as in (2.3) with coefficients h n (s 1 , s 2 ; τ ; x 1 , x 2 ) := n -2 nτ t=1 2 i=1 1 - x i n t-s i 1(t ≥ s i ), s 1 , s 2 ∈ Z. (3.34) By Proposition 2.1, with α 1 = α 2 = n, the point-wise convergence E[e iθz N,n (τ ;x 1 ,x 2 ) -1] = E[e iθQ 12 (hn(•;τ ;x 1 ,x 2 )) -1] → E[e iθz(τ ;x 1 ,x 2 ) -1] (3.35)
for any fixed x 1 , x 2 ∈ R + follows from L 2 -convergence of the kernels:

h n (•; τ ; x 1 , x 2 ) -h(•; τ ; x 1 , x 2 ) → 0, (3.36)
where

h n (s 1 , s 2 ; τ ; x 1 , x 2 ) := nh n ( ns 1 , ns 2 ; τ ; x 1 , x 2 ) = 1 n nτ t=1 2 i=1 1 - x i n t-ns i 1(t ≥ ns i ) → τ 0 2 i=1 e -x i (t-s i ) 1(t > s i )dt =: h(s 1 , s 2 ; τ ; x 1 , x 2 ) (3.37)
point-wise for any x i > 0, s i ∈ R, s i = 0, i = 1, 2, τ > 0 fixed. We also use the dominating bound

| h n (s 1 , s 2 ; τ ; x 1 , x 2 )| ≤ Ch(s 1 , s 2 ; 2τ ; x 1 , x 2 ), s 1 , s 2 ∈ R, 0 < x 1 , x 2 < n, (3.38)
with C > 0 independent of s i , x i , i = 1, 2 which follows from the definition of h n (•; τ ; x 1 , x 2 ) and the inequality 1 It remains to show the convergence of the corresponding integrals, viz.,

-x ≤ e -x , x > 0. Since h(•; 2τ ; x 1 , x 2 ) ∈ L 2 (R 2 ), ( 3
(0,n] 2 E[e iθz N,n (τ ;x 1 ,x 2 ) -1](x 1 x 2 ) β-1 dx 1 dx 2 → R 2 + E[e iθz(τ ;x 1 ,x 2 ) -1](x 1 x 2 ) β-1 dx 1 dx 2 = Φ(θ). (3.39)
From (3.31) and Ez N,n (τ ;

x 1 , x 2 ) = 0 we obtain E e iθz N,n (τ ;x 1 ,x 2 ) -1 ≤ C    1, x 1 x 2 (x 1 + x 2 ) ≤ 1, Ez 2 N,n (τ ; x 1 , x 2 ), x 1 x 2 (x 1 + x 2 ) > 1, (3.40)
where Proof of (3.30). Choose q = q N,n = log n . Let J q (θ) denote the l.h.s. of (3.30). Using the identity D⊂{1,...,q-1}:|D|≥2 i∈D w i = 1≤i<j<q w i w j i<k<j (1 + w k ) with w i = e iθy i (τ ) -1, see (3.27), we can rewrite J q (θ) = 1≤i<j<q T ij (θ), where

Ez 2 N,n (τ ; x 1 , x 2 ) = A -2 N,n E nτ t=1 Y 12 (t) 2 |a i = 1 - x i B N,n , i = 1, 2 = n -4 E nτ t=1 Y 12 (t) 2 |a i = 1 - x i n , i = 1, 2 ≤ C n 3 (x 1 /n)(x 2 /n) min n, 1 (x 1 + x 2 )/n = C x 1 x 2 min 1, 1 x 1 + x 2 , ( 3
T ij (θ) := N E (e iθy i (τ ) -1)(e iθy j (τ ) -1) exp iθ i<k<j y k (τ ) (1(a i < a j+1 ) + 1(a i > a j+1 )) (3.42) = T ij (θ) + T ij (θ). Since |J q (θ)| ≤ q 2 max 1≤i<j<q |T ij (θ)| ≤ (log n) 2 max 1≤i<j<q |T ij (θ)|, (3.30) follows from |T ij (θ)| ≤ Cn -δ , ∀ 1 ≤ i < j, (3.43)
with C, δ > 0 independent of n. Using E[y i (τ )|a k , ε j (k), k, j ∈ Z, j = i] = 0 and (3.41) we obtain

|T ij (θ)| ≤ CN E min 1, E[y 2 i (τ )|a k , k ∈ Z] 1(a i < a j+1 ) (3.44) ≤ Cn -β (0,n] 3 min 1, 1 x i x i+1 (x i + x i+1 ) (x i x i+1 x j+1 ) β-1 1(x j+1 < x i )dx i dx i+1 dx j+1 = Cn -β (0,n] 2 min 1, 1 x 1 x 2 (x 1 + x 2 ) x 2β-1 1 x β-1 2 dx 1 dx 2 ≤ Cn -β (T n + T n ),
where

T n := 0<x 1 <x 2 <n min 1, 1 x 1 x 2 2 x 2β-1 1 x β-1 2 dx 1 dx 2 , T n := 0<x 2 <x 1 <n min 1, 1 x 2 1 x 2 x 2β-1 1 x β-1 2 dx 1 dx 2 . Next, T n ≤ 1 0 x 2β-1 1 dx 1 1/ √ x 1 x 1 x β-1 2 dx 2 + x -1 1 n 1/ √ x 1 x β-3 2 dx 2 + n 1 x 2β-2 1 dx 1 n x 1 x β-3 2 dx 2 ≤ C 1 0 x 3β/2-1 1 dx 1 + n 1 x 3β-4 1 dx 1 ≤ Cn 3(β-1)∨0 (1 + 1(β = 1) log n).
Similarly,

T n = 1 0 x 2β-1 1 dx 1 x 1 0 x β-1 2 dx 2 + n 1 x 2β-1 1 dx 1 x -2 1 0 x β-1 2 dx 2 + n 1 x 2β-3 1 dx 1 x 1 x -2 1 x β-2 2 dx 2 ≤ C (log n)1(β < 1) + (log n) 2 1(β = 1) + n 3(β-1) 1(β > 1)
.

Whence, the bound in (3.43) follows for T ij (θ) with any 0 Consider the second relation in (3.22). Let L q := N -q Ñq be the number of summands in S ‡ N,n;q (τ ). Then

< δ < β ∧ (3 -2β), for 0 < β < 3/2. Since |T ij (θ)| ≤ CN E[min{1, E[y 2 j (τ )|a k , k ∈ Z]}1(a j+1 < a i )] can
A -1 N,n S ‡ N,n;q (τ ) = fdd Lq i=1 y i (τ ) and Ee iθA -1 N,n S ‡ N,n;q (τ ) -1 = L q E[e iθy 1 (τ ) -1] + |D|≥2 E i∈D e iθy i (τ ) -1 , (3.46)
where the last sum is taken over all D ⊂ {1, . . . , L q }, |D| ≥ 2. Since L q < q = o(N ) from (3.29), (3.30) we infer that the r.h.s. of (3.46) vanishes, proving (3.22), and thus completing the proof of Theorem 3.1, case (iii).

3.2 Proof of Theorem 3.1 (ii): case λ ∞ = 0, or N = o(n 2β ).

Proof of (3.29). Note the log-ch.f. of the r.h.s. in (3.5) can be written as

Φ(θ) = log Ee iθA 1/2 B(τ ) = log Ee -(θ 2 τ /2)A = -σ 0 (θ 2 τ /2) 2β/3 = -ψ(1) 2 R 2 + 1 -exp - θ 2 τ 4x 1 x 2 (x 1 +x 2 ) (x 1 x 2 ) β-1 dx 1 dx 2 (3.47)
with σ 0 > 0 given by the integral

σ 0 := ψ(1) 2 2 -2β/3 R 2 + 1 -exp - 1 x 1 x 2 (x 1 +x 2 ) (x 1 x 2 ) β-1 dx 1 dx 2 .
(3.48) Relation (3.47) follows by change of variable x i → (θ 2 τ /4)

z N,n (τ ; x 1 , x 2 ) := N -3/(4β) n -1/2 s 1 ,s 2 ∈Z ε 1 (s 1 )ε 2 (s 2 ) nτ t=1 2 i=1 1 - x i N 1/(2β) t-s i 1(t ≥ s i ).
(3.49)

Let us prove the (conditional) CLT:

z N,n (τ ; x 1 , x 2 ) → fdd (2x 1 x 2 (x 1 + x 2 )) -1/2 B(τ ), (3.50)
implying the point-wise convergence

E[1 -e iθz N,n (τ ;x 1 ,x 2 ) ] → 1 -e -θ 2 τ /(4x 1 x 2 (x 1 +x 2 )) (3.51)
of the integrands in (3.31) and (3.48), for any fixed (x 1 , x 2 ) ∈ R 2 + . As in the rest of the paper, we restrict the proof of (3.50) to one-dimensional convergence, and set τ = 1 for concreteness. Split (3.49) as z N,n (1;

x 1 , x 2 ) = z + N,n (x 1 , x 2 ) + z - N,n (x 1 , x 2 ), where z + N,n (x 1 , x 2 ) := N -3/4β n -1/2 n s 1 ,s 2 =1 ε 1 (s 1 )ε 2 (s 2 ) • • • corresponds to the sum over 1 ≤ s 1 , s 2 ≤ n alone. Thus, we shall prove that z - N,n (x 1 , x 2 ) = o p (1) and z + N,n (x 1 , x 2 ) → d N 0, 1 2x 1 x 2 (x 1 + x 2 ) . (3.52)
Arguing as in the proof of (2.29) it is easy to show that

E(z - N,n (x 1 , x 2 )) 2 ≤ C N 3/(2β) n x 1 + x 2 N 1/(2β) -2 x 1 N 1/(2β) -2 + x 2 N 1/(2β) -2 + x 1 N 1/(2β) -1 x 2 N 1/(2β) -1 = Cλ N,n (x 1 + x 2 ) -2 x -2 1 + x -2 2 + (x 1 x 2 ) -1 ,
where λ N,n → 0, implying the first relation in (3.52). To prove the second relation in (3.52) we use the martingale CLT in Hall and Heyde [START_REF] Hall | Martingale Limit Theory and Its Applications[END_REF]. (The same approach is used to prove CLT for quadratic forms in [START_REF] Bhansali | Convergence of quadratic forms with nonvanishing diagonal[END_REF].) Towards this aim, write z + N,n (x 1 , x 2 ) as a sum of zero-mean square-integrable martingale difference array

z + N,n (x 1 , x 2 ) = n k=1 Z k , Z k := ε 1 (k) k-1 s=1 f (k, s) ε 2 (s) + ε 2 (k) k-1 s=1 f (s, k) ε 1 (s) + ε 1 (k)ε 2 (k)f (k, k)
with respect to the filtration F k generated by {ε i (s), 1

≤ s ≤ k, i = 1, 2}, 0 ≤ k ≤ n, where f (s 1 , s 2 ) := N -3/(4β) n -1/2 n t=1 2 i=1 1 - x i N 1/(2β) t-s i 1(t ≥ s i ), 1 ≤ s 1 , s 2 ≤ n.
Accordingly, the second convergence in (3.52) follows from

n k=1 E[Z 2 k |F k-1 ] → p 1 2x 1 x 2 (x 1 + x 2 )
and

n k=1 E[Z 2 k 1(|Z k | > )] → 0 for any > 0. (3.53) Note the conditional variance v 2 k := E[Z 2 k |F k-1 ] = k-1 s=1 f (k, s)ε 2 (s) 2 + k-1 s=1 f (s, k)ε 1 (s) 2 + f 2 (k, k),
where

n k=1 EZ 2 k = n k=1 Ev 2 k = n s 1 ,s 2 =1 f 2 (s 1 , s 2 ) = E(z + N,n (x 1 , x 2 )) 2 → 1 2x 1 x 2 (x 1 + x 2 ) (3.54)
is a direct consequence of the asymptotics in (2.27), where a i = 1-x 1 /N 1/(2β) , a j = 1-x 2 /N 1/(2β) . Therefore the first relation in (3.53) follows from (3.54) and

R n := n k=1 (v 2 k -Ev 2 k ) = o p (1). (3.55)
To show (3.55) we split R n = R n + R n into the sum of 'diagonal' and 'off-diagonal' parts, viz.,

R n := 2 i=1 1≤s<n c i (s)(ε 2 i (s) -1), R n := 2 i=1 1≤s 1 ,s 2 <n,s 1 =s 2 c i (s 1 , s 2 )ε i (s 1 )ε i (s 2 ),
where

c 1 (s) := s<k≤n f 2 (s, k), c 2 (s) := s<k≤n f 2 (k, s), c 1 (s 1 , s 2 ) := s 1 ∨s 2 <k≤n f (s 1 , k)f (s 2 , k), c 2 (s 1 , s 2 ) := s 1 ∨s 2 <k≤n f (k, s 1 )f (k, s 2 ).
Using the elementary bound for 1 ≤ s 1 , s 2 ≤ n:

n t=1 2 i=1 a t-s i i 1(t ≥ s i ) ≤ a s 1 -s 2 2 1(1 ≤ s 2 ≤ s 1 ) + a s 2 -s 1 1 1(1 ≤ s 1 ≤ s 2 ) S(a 1 , a 2 ), S(a 1 , a 2 ) := ∞ t=0 (a 1 a 2 ) t = (1 -a 1 a 2 ) -1 ≤ 2(2 -a 1 -a 2 ) -1 , we obtain |c i (s)| ≤ Cn -1 x -1 i (x 1 + x 2 ) -2 , n s 1 ,s 2 =1 c 2 i (s 1 , s 2 ) ≤ Cλ N,n x -3 i (x 1 + x 2 ) -4 , i = 1, 2. (3.56)
By (3.56), for 1 < p < 2 and x 1 , x 2 > 0 fixed Now we return to the proof of (3.29), whose both sides are written as respective integrals (3.31) and (3.47). Due to the convergence of the integrands (see (3.51)), it suffices to justify the passage to the limit using a dominated convergence theorem argument. The dominating function independent of N, n is obtained from (3.31) and Ez N,n (τ ; x 1 , x 2 ) = 0 and from (3.40), (3.41), (2.8) similarly as in the case λ ∞ ∈ (0, ∞) above. This proves (3.29).

E|R n | p ≤ C 2 i=1 n-1 s=1 |c i (s)| p ≤ Cn -(p-1) = o(1), (3.57) E|R n | 2 ≤ 2 i=1 n s 1 ,s 2 =1 c 2 i (s 1 , s 2 ) ≤ Cλ N,n = o(1), (3.58
:= n k=1 E[|Z k | 2p ] = o(1) for the same 1 < p ≤ 2, where E|Z k | 2p ≤ C E| k-1 s=1 f (k, s) ε 2 (s)| 2p + E| k-1 s=1 f (s, k) ε 1 (s)| 2p + |f (k, k)| 2p ≤ C ( k-1 s=1 f 2 (k, s)) p + ( k-1 s=1 f 2 (s, k)) p + |f (k, k)| 2p
Proofs of (3.30) and (3.22) are completely analogous to those in the case λ ∞ ∈ (0, ∞) except that we now choose q = log N and replace n in (3.43) and elsewhere in the proof of (3.30) and (3.22), case λ ∞ ∈ (0, ∞), by N 1/2β . This ends the proof of Theorem 3.1, case (ii).

Proof of

Theorem 3.1 (i): case λ ∞ = ∞, or n = o(N 1/(2β) ) Case 1 < β < 3/2. Proof of (3.29). In this case, Φ(θ) := -σ 2 ∞ τ 2(2-β) θ 2 /2, B N,n = n and A N,n = n 2 λ β N,n = n 2-β N 1/2
. Rewrite the l.h.s. of (3.29) as

N Ee iθy 1 (τ ) -1 = [0,n) 2 EΛ N,n (θ; τ ; x 1 , x 2 )(x 1 x 2 ) β-1 dx 1 dx 2 , where (3.59) Λ N,n (θ; τ ; x 1 , x 2 ) := λ 2β N,n e iθλ -β N,n zN,n (τ ;x 1 ,x 2 ) -1 -iθλ -β N,n zN,n (τ ; x 1 , x 2 )
and where zN,n (τ ; x 1 , x 2 ) is defined as in (3.32) with A N,n replaced by ÃN,n := n 2 = A N,n /λ β N,n . As shown in the proof of Case (iii) (the 'intermediate limit'), for any

x 1 , x 2 > 0 zN,n (τ ; x 1 , x 2 ) → d z(τ ; x 1 , x 2 ) and Ez 2 N,n (τ ; x 1 , x 2 ) → Ez 2 (τ ; x 1 , x 2 ), ( 3
G N,n (τ ; x 1 , x 2 ) → -1 2 Ez 2 (τ ; x 1 , x 2 ), ∀ (x 1 , x 2 ) ∈ R 2 + . ( 3
|G N,n (τ ; x 1 , x 2 )| ≤ CEz 2 N,n (τ ; x 1 , x 2 ) ≤ C x 1 x 2 min 1, 1 x 1 +x 2 =: Ḡ(x 1 , x 2 ), (3.62)
see (3.41), and integrability of Ḡ, see (2.9).

Proof of (3.30) is similar to that in case (iii) 0 < λ ∞ < ∞ above with q = log n . It suffices to check the bound (3.43) for T ij (θ) = T ij (θ) + T ij (θ) given in (3.42). By the same argument as in (3.44), we obtain

|T ij (θ)| ≤ CN E[y 2 i (τ )1(a i < a j+1 )].
The bound on Ez 2 N,n (τ ; x 1 , x 2 ) in (3.62) further implies

|T ij (θ)| ≤ Cn -β (0,n] 3 1 x 1 x 2 min 1, 1 x 1 + x 2 (x 1 x 2 x 3 ) β-1 1(x 3 < x 1 )dx 1 dx 2 dx 3 ≤ Cn -β (T n + T n ),
where

T n := n 0 min 1, 1 x 1 x 2β-2 1 dx 1 x 1 0 x β-2 2 dx 2 = C 1 0 x 3β-3 1 dx 1 + n 1 x 3β-4 1 dx 1 ≤ Cn 3β-3
and

T n := n 0 min 1, 1 x 2 x β-2 2 dx 2 x 2 0 x 2β-2 1 dx 1 = C 1 0 x 3β-3 2 dx 2 + n 1 x 3β-4 2 dx 2 ≤ Cn 3β-3 . Then |T ij (θ)| ≤ CN E[y 2 j (τ )1(a i > a j+1 )
] can be handled in the same way. Whence, the bound in (3.43) follows with any 0 < δ < 3 -2β, for 1 < β < 3/2. This proves (3.30). Proof of (3.22) using Ñq /N → 0 and L q = N -q Ñq < q = o(N ) is completely analogous to that in case (iii) 0 < λ ∞ < ∞. This completes the proof of Theorem 3.1, case (i) for 1 < β < 3/2. Case 0 < β < 1. Proof of (3.29). In the rest of this proof, write λ ≡ λ N,n = N 1/(2β) /n → ∞ for brevity. Also denote λ := λ(log λ)

1/2β , log λ / log λ → 1. Let B N,n := λ n, then z N,n (τ ; x 1 , x 2 ) := 1 λ n 2 s 1 ,s 2 ∈Z ε 1 (s 1 )ε 2 (s 2 ) nτ t=1 2 i=1 1 - x i λ n t-s i 1(t ≥ s i ). (3.63)
Split the r.h.s. of (3.29) as follows:

N E e iθy 1 (τ ) -1 = 1 log λ (0,λ n] 2 1(1 < x 1 + x 2 < λ) + 1(x 1 + x 2 > λ) + 1(x 1 + x 2 < 1) ×E e iθz N,n (τ ;x 1 ,x 2 ) -1 (x 1 x 2 ) β-1 dx 1 dx 2 =: 3 i=1 L i .
Here, L 1 is the main term and L i , i = 2, 3 are remainders. Indeed,

|L 3 | = O(1/ log λ) = o(1). To estimate L 2 we need the bound Ez 2 N,n (τ ; x 1 , x 2 ) ≤ C x 1 x 2 min 1, λ x 1 + x 2 , ( 3
|L 2 | ≤ C log λ x 1 +x 2 >λ min 1, λ x 1 x 2 (x 1 + x 2 ) (x 1 x 2 ) β-1 dx 1 dx 2 = C log λ (J λ + J λ ), (3.65)
where, by change of variables:

x 1 + x 2 = y, x 1 = yz, J λ := x 1 +x 2 >λ 1(x 1 x 2 (x 1 + x 2 ) < λ )(x 1 x 2 ) β-1 dx 1 dx 2 = ∞ λ 1 0 1(y 3 z(1 -z) < λ )y 2β-1 (z(1 -z)) β-1 dzdy ≤ C ∞ λ y 2β-1 dy 1/2 0 z β-1 1(y 3 z < 2λ )dz ≤ C(λ ) β ∞ λ y -β-1 dy = C(log λ) 1/2
since 0 < β < 1. Similarly,

J λ := λ x 1 +x 2 >λ 1(x 1 x 2 (x 1 + x 2 ) > λ )(x 1 + x 2 ) -1 (x 1 x 2 ) β-2 dx 1 dx 2 ≤ Cλ ∞ λ y 2β-4 dy 1/2 0 z β-2 1(y 3 z > λ )dz ≤ C(log λ) 1/2 .
This proves

|L 2 | = O(1/ log λ) = o(1).
Consider the main term L 1 . Although Ee iθz N,n (τ ;x 1 ,x 2 ) and hence the integrand in L 1 point-wise converge for any (x 1 , x 2 ) ∈ R 2 + , see below, this fact is not very useful since the contribution to the limit of L 1 from bounded x i 's is negligible due to the presence of the factor 1/ log λ → 0 in front of this integral. It turns out that the main (non-negligible) contribution to this integral comes from unbounded x 1 , x 2 with

x 1 /x 2 + x 2 /x 1 → ∞ and x 1 x 2 → z ∈ R + .
To see this, by change of variables y = x 1 + x 2 , x 1 = yw and then w = z/y 2 we rewrite

L 1 = 1 log λ λ 1 V N,n (θ; y) dy y , (3.66)
where

V N,n (θ; y) := 2 y 2 /2 0 E exp iθz N,n (τ ; z y , y 1 - z y 2 -1 z β-1 1 - z y 2 β-1 dz. (3.67)
In view of L i = o(1), i = 2, 3 relation (3.29) follows from representation (3.66) and the existence of the limit:

lim y→∞,y=O(λ) V N,n (θ; y) = V (θ) := -k ∞ |θ| 2θ τ 2β , (3.68)
where the constant k ∞ > 0 is defined below in (3.71). More precisely, (3.68) says that for any > 0 there exists K > 0 such that for any N, n, y To prove (3.69), rewrite V (θ) of (3.68) as the integral

≥ K satisfying y ≤ λ, λ ≥ K |V N,n (θ; y) -V (θ)| < . (3.69) To show that (3.69) implies L 1 → V (θ) it suffices to split L 1 -V (θ) = (log λ) -1 λ K (V N,n (θ; y) -V (θ)) dy y + (log λ) -1 K 1 (V N,n (θ; y) -V (θ))
V (θ) = 2 ∞ 0 z β-1 E(e iθτ Z 1 Z 2 /(2 √ z) -1)dz = -2E ∞ 0 z β-1 (1 -e -θ 2 τ 2 Z 2 1 /(8z) )dz = -k ∞ |θ| 2β τ 2β (3.70)
with Z 1 , Z 2 ∼ N (0, 1) independent normals and

k ∞ = 2E ∞ 0 z β-1 (1 -e -Z 2 1 /(8z) )dz = 2 1-3β E|Z 1 | 2β ∞ 0 z β-1 (1 -e -1/z )dz = 2 1-2β Γ(β + 1 2 )Γ(1 -β)/( √ πβ). (3.71) Let Λ N,n (z; y) := E exp iθz N,n (τ ; z y , y 1 -z y 2 -1 , Λ(z) := E[e iθτ Z 1 Z 2 /(2 √ z) -1]
) = Q 12 (h α 1 ,α 2 (•; τ ; z)) with h α 1 ,α 2 (s 1 , s 2 ; τ ; z) := y zy 1 √ α 1 α 2 1 n nτ t=1 2 i=1 1 - 1 α i t-s i 1(t ≥ s i ), s 1 , s 2 ∈ Z, (3.74
)

α 1 := λ ny/z, α 2 := λ n/y . If n, α 1 , α 2 , y, y → ∞ so that y/y → 1 and n = o(α i ), i = 1, 2, (3.75) then h (α 1 ,α 2 ) α 1 ,α 2 (s 1 , s 2 ; τ ; z) := √ α 1 α 2 h α 1 ,α 2 ( α 1 s 1 , α 2 s 2 ; τ ; z) = y zy 1 n nτ t=1 2 i=1 1 - 1 α i t-α i s i 1(t ≥ α i s i ) → τ √ z 2 i=1 e s i 1(s i < 0) =: h(s 1 , s 2 ; τ ; z) (3.76)
point-wise for any τ > 0, z > 0, s i ∈ R, s i = 0, i = 1, 2 fixed. Moreover, under the same conditions (3.75), h Proof of (3.30). For T ij (θ) defined by (3.42) let us prove (3.43). Denote N λ := (N log λ) 1/2β . Similarly to (3.44) we have that

(α 1 ,α 2 ) α 1 ,α 2 (•; τ ; z) -h(•; τ ; z) → 0, implying the convergence Q 12 (h α 1 ,α 2 (•; τ ; z)) → d I 12 (h(•; τ ; z)) = d τ Z 1 Z 2 /(2 √ z), Z i ∼ N (0, 1), i = 1,
|T ij (θ)| ≤ C N 1/2 (log λ) 3/2 (0,N λ ] 3 min 1, Ez 2 N,n (τ ; x 1 , x 2 ) (x 1 x 2 x 3 ) β-1 1(x 3 < x 1 )dx 1 dx 2 dx 3
with z N,n (τ ; x 1 , x 2 ) defined by (3.63). Whence using (3.64) similarly as in the proof of case (i) we obtain

|T ij (θ)| ≤ C N 1/2 (log λ) 3/2 (0,N λ ] 2 min 1, 1 x 1 x 2 min 1, λ x 1 +x 2 x 2β-1 1 x β-1 2 dx 1 dx 2 = C N 1/2 (log λ) 3/2 3 i=1 T λ,i ,
where

T λ,1 := (0,N λ ] 2 1(x 1 + x 2 < λ ) min 1, 1 x 1 x 2 x 2β-1 1 x β-1 2 dx 1 dx 2 , T λ,2 := (0,N λ ] 2 1(x 1 x 2 (x 1 + x 2 ) < λ , x 1 + x 2 > λ )x 2β-1 1 x β-1 2 dx 1 dx 2 , T λ,3 := λ (0,N λ ] 2 1(x 1 x 2 (x 1 + x 2 ) > λ , x 1 + x 2 > λ )x 2β-2 1 x β-2 2 dx 1 dx 2 /(x 1 + x 2 ). By changing variables x 1 , x 2 as in (3.66)-(3.67) we get T λ,1 ≤ C λ 0 y β-1 dy ≤ C(λ ) β . Also, similarly to the estimation of J λ , J λ , following (3.65) we obtain T λ,2 + T λ,3 ≤ C(λ ) β 2N λ λ y -1 dy ≤ C(λ ) β log(N λ /λ ). Hence, we conclude that |T ij (θ)| ≤ C(λ ) β log(N λ /λ ) N 1/2 (log λ) 3/2 ≤ C log n n β log λ ,
proving (3.43) with any 0 < δ < β. This proves (3.30). We omit the proof of (3.22) which is completely similar to that in case (iii) and elsewhere. This completes the proof of Theorem 3.1 for (t, s) = (0, 1).

Proof of Theorem 3.1 in the general case (t, s) ∈ Z 2 , s ≥ 1. Similarly to (3.17) we decompose S t,s N,n (τ ) in (3.1) as S t,s N,n (τ ) = S t,s N,n;q (τ ) + S t,s; † N,n;q (τ ) + S t,s; ‡ N,n;q (τ ),

where the main term

S t,s N,n;q (τ ) := Ñq k=1 Y t,s k,n;q (τ ), Y t,s k,n;q (τ ) := (k-1)q<i≤kq-s nτ u=1 X i (u)X i+s (u + t) (3.78)
is a sum of independent Ñq = N/q blocks of size q -s = q N,n -s → ∞, and

S t,s; † N,n;q (τ ) := Ñq k=1 kq-s<i≤kq nτ u=1 X i (u)X i+s (u + t), S t,s; ‡ N,n;q (τ ) := q Ñq<i≤N nτ u=1 X i (u)X i+s (u + t)
are remainder terms. The proof of (3.29)-(3.30) for

A -1 N,n Y t,s 1,n;q (τ ) = q-s i=1 y t,s i (τ ), y t,s i (τ ) := A -1 N,n nτ u=1 X i (u)X i+s (u + t)
is completely analogous since the distribution of y t,s i (τ ) does not depend on t and s = 0.

Proof of Theorem 3.2

The proof uses the following result of [START_REF] Pilipauskaitė | Scaling transition for nonlinear random fields with long-range dependence[END_REF].

Lemma 3.1. ([23], Lemma 7.1) Let {ξ ni , 1 ≤ i ≤ N n }, n ≥ 1,
be a triangular array of m-dependent r.v.s with zero mean and finite variance. Assume that: (L1)

ξ ni , 1 ≤ i ≤ N n , are identically distributed for any n ≥ 1, (L2) ξ n1 → d ξ, Eξ 2 n1 → Eξ 2 < ∞ for some r.v. ξ and (L3) var( Nn i=1 ξ ni ) ∼ σ 2 N n , σ 2 > 0. Then N -1/2 n Nn i=1 ξ ni → d N (0, σ 2 ).
For notational simplicity, we consider only one-dimensional convergence at

τ > 0. Let (N n) -1/2 S t,s N,n (τ ) = N -1/2 N i=1 ξ ni , where ξ ni := n -1/2 nτ u=1 X i (u)X i+s (u + t), 1 ≤ i ≤ N are |s|-dependent
, identically distributed random variables with zero mean and finite variance. Since

ξ ni , 1 ≤ i ≤ N are uncorrelated, it follows that E( N i=1 ξ ni ) 2 = N Eξ 2 n1 , where ξ n1 = d ξ n := n -1/2 nτ u=1 X 1 (u)X 2 (u). Proposition 2.4 implies E[ξ 2 n |a 1 , a 2 ] ∼ τ A 12
, and so Eξ 2 n ∼ τ σ 2 , where

σ 2 := EA 12 < ∞. It remains to show that ξ n → d √ A 12 B(τ )
, where A 12 is independent of B(τ ). This follows from the martingale CLT similarly to (3.50). By the lemma above, we conclude that (N n) -1/2 S t,s N n (τ ) → d σB(τ ). Theorem 3.2 is proved.

4 Asymptotic distribution of temporal (iso-sectional) sample covariances

The limit distribution of iso-sectional sample covariances γ N,n (t, 0) in (1.5) and the corresponding partial sums process S t,0 N,n (τ ) of (3.1) is obtained similarly as in the cross-sectional case, with certain differences which are discussed below. Since the conditional expectation E

[S t,0 N,n (τ )|a 1 , • • • , a N ] =: T t,0 N,n (τ ) = 0, a natural decomposition is S t,0 N,n (τ ) = S t,0 N,n (τ ) + T t,0 N,n (τ ), (4.1)
where S t,0 N,n (τ ) := S t,0 N,n (τ ) -T t,0 N,n (τ ) is the conditionally centered term with E[ S t,0 N,n (τ )|a 1 , • • • , a N ] = 0, and

T t,0 N,n (τ ) := nτ N i=1 a t i /(1 -a 2 i ), t ≥ 0, (4.2)
is proportional to a sum of i.i.d. r.v.s a t i /(1 -a 2 i ), 1 ≤ i ≤ N with regularly decaying tail distribution function

P a t /(1 -a 2 ) > x) ∼ P(a > 1 - 1 2x ) ∼ c a x -β , x → ∞, c a := ψ(1)/2 β β, see condition (1.
2). Accordingly, the limit distribution of appropriately normalized and centered term T t,0 N,n (τ ) does not depend on t and can be found from the classical CLT and turns out to be a (β ∧ 2)-stable line, under normalization nN 1/(β∧2) (β = 2). The other term, S t,0 N,n (τ ), in (4.1), is a sum of mutually independent partial sums processes Y t,0 i,n (τ

) := nτ u=1 (X i (u)X i (u + t) -E[X i (u)X i (u + t)|a i ]), 1 ≤ i ≤ N with conditional variance var[Y t,0 i,n (1)|a i ] ∼ nA t,0 ii , n → ∞, where A t,0 ii := 1+a 2 i 1-a 2 i 1+a 2|t| i (1-a 2 i ) 2 + a 2|t| i (2|t|+cum 4 ) 1-a 4 i .
The proof of the last fact follows similarly to that of (2.28) and is omitted. As a i ↑ 1, A t,0 ii ∼ 1/(2(1-a i ) 3 ) and the limit distribution of S t,0 N,n (τ ) can be shown to exhibit a trichotomy on the interval 0 < β < 3 depending on the limit λ * ∞ in (4.3). It turns out that for β > 2 the asymptotically Gaussian term T t,0 N,n (τ ) dominates S t,0 N,n (τ ) in all cases of λ * ∞ , while in the interval 0 < β < 2 T t,0 N,n (τ ) and S t,0 N,n (τ ) have the same convergence rate. Somewhat surprisingly, the limit distribution of S t,0 N,n (τ ) is a β-stable line in both cases λ * ∞ = ∞ and λ * ∞ = 0 with different scale parameters of the random slope coefficient of this line. Rigorous description of the above limit results is given in the following Theorems 4.1 and 4.2. The proofs of these theorems are similar and actually simpler than the corresponding Theorems 3.1 and 3.2 dealing with non-horizontal sample covariances, due to the fact that S t,0 N,n (τ ) is a sum of row-independent summands contrary to S t,s N,n (τ ), s = 0. Because of this, we omit some details of the proof of Theorems 4.1 and 4.2. We also omit the more delicate cases β = 1 and β = 2 where the limit results may require a change of normalization or additional centering.

with 0 < β < 2, β = 1. Let N, n → ∞ so that λ * N,n := N 1/β n → λ * ∞ ∈ [0, ∞]. (4.3)
In addition, assume Eε 4 (0) < ∞. Then the following statements (i)-(iii) hold for S t,0 N,n (τ ), t ∈ Z in (3.1) depending on λ * ∞ in (4.3).

(i) Let λ * ∞ = ∞. Then n -1 N -1/β S t,0 N,n (τ ) -ES t,0 N,n (τ )1(1 < β < 2) → fdd τ V * β , (4.4)
where V * β is a completely asymmetric β-stable r.v. with characteristic function in (4.7) below.

(ii) Let λ * ∞ = 0. Then

n -1 N -1/β S t,0 N,n (τ ) -ES t,0 N,n (τ )1(1 < β < 2) → fdd τ V + β , (4.5)
where V + β is a completely asymmetric β-stable r.v. with characteristic function in (4.8) below.

(iii) Let 0 < λ * ∞ < ∞. Then n -1 N -1/β S t,0 N,n (τ ) -ES t,0 N,n (τ )1(1 < β < 2) → fdd λ * ∞ Z * β (τ /λ * ∞ ), (4.6)
where Z * β is the 'diagonal intermediate' process in (2.24).

Remark 4.1. The r.v.s V * β and V + β in (4.4) and (4.5) have respective stochastic integral representations [START_REF] Beran | From short to long memory: Aggregation and estimation[END_REF]. The fact that both V * β and V + β have completely asymmetric β-stable distribution follows from their ch.f.s:

V * β = R + ×C(R) 0 -∞ e xs dB(s) 2 d(M * β -EM * β 1(1 < β < 2)), V + β = R + ×C(R) (2x) -1 d(M * β -EM * β 1(1 < β < 2)) w.r.t. Poisson random measure M * β in (2.21). Note 0 -∞ e xs dB(s) = law Z/ √ 2x, Z ∼ N (0,
Ee iθV * β = exp ψ(1) ∞ 0 E e iθZ 2 /(2x) -1 -i(θZ 2 /(2x))1(1 < β < 2) x β-1 dx = exp -c * β |θ| β (1 -i sign(θ) tan(πβ/2)) , (4.7)
Ee iθV + β = exp ψ(1) ∞ 0 e iθ/(2x) -1 -i(θ/(2x))1(1 < β < 2) x β-1 dx = exp -c + β |θ| β (1 -i sign(θ) tan(πβ/2)) , θ ∈ R, (4.8)
where

c + β := ψ(1)Γ(2 -β) cos(πβ/2) 2 β β(1 -β) , c * β := c + β E|Z| 2β (4.9)
with E|Z| 2β = 2 β Γ(β + 1/2)/ √ π = 1 unless β = 1, implying that V * β and V + β have different distributions.

Theorem 4.2. Let the mixing distribution satisfy condition (1.2) with β > 2. In addition, assume Eε 4 (0) < ∞. Then for any t ∈ Z, as N, n → ∞ in arbitrary way,

n -1 N -1/2 S t,0 N,n (τ ) -ES t,0 N,n (τ ) → fdd τ σ * t Z, (4.10)
where Z ∼ N (0, 1) and (σ * t ) 2 := var(a |t| /(1 -a 2 )).

Remark 4.2. If β < 1, then γ(t, 0) is undefined for any t ∈ Z. Using the convention γ(t, 0)1(1 < β < 2) := 0 if β < 1 and γ(t, 0) if β > 1.

Corollary 4.1. (i) Let the conditions of Theorem 4.1 (i) be satisfied. Then for any t ∈ Z

N 1-1/β ( γ N,n (t, 0) -γ(t, 0)1(1 < β < 2)) → d V * β .
(ii) Let the conditions of Theorem 4.1 (ii) be satisfied. Then for any t ∈ Z

N 1-1/β ( γ N,n (t, 0) -γ(t, 0)1(1 < β < 2)) → d V + β .
(iii) Let the conditions of Theorem 4.1 (iii) be satisfied. Then for any t ∈ Z

N 1-1/β ( γ N,n (t, 0) -γ(t, 0)1(1 < β < 2)) → d λ * ∞ Z * β (1/λ * ∞ ).
(iv) Let the conditions of Theorem 4.2 be satisfied. Then for any t ∈ Z

N 1/2 ( γ N,n (t, 0) -γ(t, 0)) → d σ * t Z, Z ∼ N (0, 1).
Proof of Theorem 4.1. Let t ≥ 0 and

y t,0 (τ ) := 1 nN 1/β nτ u=1 (X(u)X(u + t) -EX(u)X(u + t)1(1 < β < 2)). (4.11)
It suffices to prove that

Φ t,0 N,n (θ) → Φ * (θ), as N, n → ∞, λ * N,n → λ * ∞ , ∀θ ∈ R, (4.12)
where, using Ey t,0 (τ )1(1 < β < 2) = 0, Φ t,0 N,n (θ) := N E e iθy t,0 (τ ) -1 -iθy t,0 (τ )1(1

< β < 2) , Φ * (θ) := log Ee iθS * β (τ ) , (4.13)
and S * β (τ ) denotes the limit process in (4.4)-(4.6). Similarly to (3.31), Φ t,0 N,n (θ) = ψ(1)

(0,1/N 1/β ] E e iθz t,0 N,n (τ ;x) -1 -iθz t,0 N,n (τ ; x)1(1 < β < 2) x β-1 dx, (4.14)
where z t,0 N,n (τ ; x) := y t,0 (τ )| a=1-x/N 1/β . Next we decompose y t,0 (τ ) = y * (τ ) + y + (τ ), where

y * (τ ) := 1 nN 1/β nτ u=1 (X(u)X(u + t) -E[X(u)X(u + t)|a]), y + (τ ) := nτ nN 1/β (E[X(0)X(t)|a] -E[X(0)X(t)1(1 < β < 2)]) = nτ nN 1/β a t 1 -a 2 -E a t 1(1 < β < 2) 1 -a 2 .
Accordingly, we decompose z t,0 N,n (τ ; x) = z * N,n (τ ; x) + z + N,n (τ ; x), where

z * N,n (τ ; x) := 1 nN 1/β s 1 ,s 2 ∈Z ε(s 1 )ε(s 2 ) nτ u=1 1 -x N 1/β 2u+t-s 1 -s 2 1(u ≥ s 1 , u + t ≥ s 2 ), (4.15)
z + N,n (τ ; x) := nτ nN 1/β (1 -xN -1/β ) t 1 -(1 -xN -1/β ) 2 -E a t 1(1 < β < 2) 1 -a 2 ,
where ε(s 1 )ε(s 2 ) := ε(s 1 )ε(s 2 ) -Eε(s 1 )ε(s 2 ).

Proof of (4.12), case 0 < λ * ∞ < ∞. We have

Φ * (θ) = ψ(1) ∞ 0 E e iθλ * ∞ z * (τ /λ * ∞ ;x) -1 -iθλ * ∞ z * (τ /λ * ∞ ; x)1(1 < β < 2) x β-1 dx, (4.16)
where the last expectation is taken w.r.t. the Wiener measure P B . Similarly as in the proof of (3.29) we prove the point-wise convergence of the integrands in (4.14) and (4.16): for any x > 0

Λ t,0 N,n (θ; x) := E e iθz t,0 N,n (τ ;x) -1 -iθz t,0 N,n (τ ; x)1(1 < β < 2) (4.17) → E e iθλ * ∞ z * (τ /λ * ∞ ;x) -1 -iθλ * ∞ z * (τ /λ * ∞ ; x)1(1 < β < 2) .
The proof of (4.17) using Proposition 2.1 is very similar to that of (3.35) and we omit the details. Using (4.17) and the dominated convergence theorem we can prove the convergence of integrals, or (4.12). The application of the dominated convergence theorem is guaranteed by the dominating bound

|Λ t,0 N,n (θ; x)| ≤ C(1 ∧ (1/x)){1(0 < β < 1) + (1/x)1(1 < β < 2)}, (4.18)
which is a consequence of |z + N,n (τ ;

x)| ≤ C/x, E(z * N,n (τ ; x)) 2 ≤ Cx -2 , see (2.29). Particularly, for 0 < β < 1 we get |Λ t,0 N,n (θ; x)| ≤ 2 and |Λ t,0 N,n (θ; x)| ≤ E(|z * N,n (τ ; x)| + |z + N,n (τ ; x)|) ≤ C( E|z * N,n (τ ; x)| 2 + (1/x)) ≤ C/
x, hence (4.18) follows. For 1 < β < 2 (4.18) follows similarly. This proves (4.12) for 0 < λ * ∞ < ∞.

Proof of (4.12), case λ * ∞ = 0. In this case

Φ * (θ) = ψ(1) R + e iθ(τ /(2x)) -1 -iθ(τ /(2x))1(1 < β < 2) x β-
Λ t,0 N,n (θ; x) → e iθτ /(2x) -1 -iθ(τ /(2x))1(1 < β < 2)
for any x > 0 similarly as in (4.17). Finally, the use of the dominating bound in (4.18) which is also valid in this case completes the proof of (4.12) for λ * ∞ = 0.

Proof of (4.12), case λ * ∞ = ∞. In this case,

Φ * (θ) = ψ(1) R + E e iθ(τ Z 2 /(2x)) -1 -iθ(τ Z 2 /(2x))1(1 < β < 2) x β-1 dx, ( 4
= α 2 ≡ α := N 1/β . Note h (α,α) (τ ; x; s 1 , s 2 ) = n -1 nτ u=1 (1 -x/N 1/β ) u-N 1/β s 1 (1 - x/N 1/β ) t+u-N 1/β s 2 1(u ≥ N 1/β s 1 , u + t ≥ N 1/β s 2 ) → g(s 1 , s 2 ) := τ e x(s 1 +s 2 ) 1(s 1 ∨ s 2 ≤ 0) point-wise a.e. in (s 1 , s 2 ) ∈ R 2 and also in L 2 (R 2 ). Then conclude z * N,n (τ ; x) → d I 11 (g) = d R 2 g(s 1 , s 2 )dB(s 1 )dB(s 2 ) = d τ 0 -∞ e sx dB(s) 2 -E 0 -∞ e sx dB(s) 2 = d τ (Z 2 -1
)/(2x) for any x > 0, where Z ∼ N (0, 1). On the other hand, z + N,n (τ ; x) → τ /2x and therefore

Λ t,0 N,n (θ; x) → E e iθτ Z 2 /(2x) -1 -iθ(τ Z 2 /(2x))1(1 < β < 2)
for any x > 0, proving the point-wise convergence of the integrands in (4.14) and (4.19). The remaining details are similar as in the previous cases and omitted. This ends the proof of Theorem 4.1.

Proof of Theorem 4.2. Consider the decomposition in (4.1), where n

-1 T t,0 N,n (τ ) = ( nτ /n) N i=1 a t i /(1 -a 2 i ) is a sum of i.i.d. r.v.s with finite variance (σ * t ) 2 = var(a |t| /(1 -a 2 )
) and therefore

n -1 N -1/2 T t,0 N,n (τ ) -ET t,0 N,n (τ ) → fdd τ σ * t Z
holds by the classical CLT as N, n → ∞ in arbitrary way and where Z ∼ N (0, 1). Hence, the statement of the theorem follows from S t,0 N,n (1) = o p (nN

) = N Evar[ n u=1 X(u)X(u + t)|a] ≤ CN n 2 E (1 -a) -2 min{1, (n(1 -a)) -1 } (1)
, where the last expectation vanishes as n → ∞, due to E(1 -a) -2 < ∞. Theorem 4.2 is proved. To illustrate our results, we use a 2 ∼ Beta(α, β), α, β > 0, as in [START_REF] Granger | Long memory relationship and the aggregation of dynamic models[END_REF]. Then condition (1.2) holds with the same β and we can explicitly compute parameters of the limit distributions in cases (i), (ii), (iv) of Corollary 4.1. Figure 1 shows the density of the corresponding limiting random variables for α = 2, β = 1.5, 2.5 and t = 0. We also plot the kernel density estimates constructed using 1000 RCAR(1) panels with N = 5000, n = 100, 5000, ε(0) ∼ N (0, 1). More specifically, we use a random sample of N 1/β ( γ N,n (0, 0) -γ(0, 0)) if β = 1.5 and N 1/2 ( γ N,n (0, 0) -γ(0, 0)) if β = 2.5. On the l.h.s. we can see that the empirical distribution of γ N,n (0, 0) is different for n = 100, 5000, whereas on the r.h.s. both kernel density estimates are quite close to the limiting normal density.

In the finite variance case β > 1, Corollary 4.1 can be used for statistical inference about the covariance γ(t, 0) = γ(t) in (1.3), provided parameters of the limit distributions are consistently estimated. Denote by Then for any t ∈ Z

F * β,ψ (x) := P(V * β ≤ x), F + β,ψ (x) := P(V + β ≤ x), x ∈ R, (4.20
sup x∈R P N 1-1/ βN,n ( γ N,n (t, 0) -γ(t)) ≤ x -F + βN,n , ψN,n (x) = o p (1). ( 4.24)
(iii) Let the conditions of Theorem 4.2 be satisfied, β > 2, and σ2 N,n,t be an estimator as in (4.22). Then for any t ∈ Z Remark 4.3. Using Corollary 4.2 we can construct asymptotic confidence intervals for γ(t), as follows. For α ∈ (0, 1) denote by q β,ψ (α) the α-quantile of the c.d.f. F * β,ψ in (4.20). Then, since α = F * βN,n , ψN,n (q βN,n , ψN,n (α)) a.s., P(N 1-1/ βN,n ( γ N,n (t, 0) -γ(t)) ≤ q βN,n , ψN,n (α)) -α = o p (1) follows from Hence,

sup x∈R P N σ2 N,n,t 1/2 ( γ N,n (t, 0) -γ(t)) ≤ x -P(Z ≤ x) = o p (1), Z ∼ N (0, 1). (4.25) Proof. Consider (4.23). Write N 1-1/ βN,n ( γ N,n (t, 0) -γ(t)) = N 1-1/β ( γ N,n (t, 0) -γ(t)) + ξ N,n , where ξ N,n := (N (1/β)-(1/ βN,n ) -1)N 1-1/β ( γ N,n (t, 0) -γ(t)) = o p (
|P(N 1-1/ βN,n ( γ N,n (t, 0) -γ(t)) ≤ x) -F * β,ψ (x)| → 0. Relation sup x∈R |F * β,ψ (x) -F * βN,n , ψN,n (
J β ≤ C R 2 + 1(x 1 x 2 (x 1 + x 2 ) > 1)(x 1 + x 2 ) -1 (x 1 x 2 ) β-2 dx 1 dx 2 ≤ C R 2 + 1(x 2 > x 1 , x 1 x 2 2 > 1)x β-2 1 x β-3 2 dx 1 dx 2 = C 1 0 x β-2 1 dx 1 ∞ x -1/2 1 x β-3 2 dx 2 + ∞ 1 x β-2 1 dx 1 ∞ x 1 x β-3 2 dx 2 < ∞ if 0 < β < 3/2.
The remaining facts in (i) are easy and we omit the details.

(ii) Similarly as in ( [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF], proof of Proposition 3.1(ii)) it suffices to show for any 0 < p < 2β that where

∞ > J p,β (τ ) :=    R 2 + E|z(τ ; x 1 , x 2 )| p (x 1 x 2 ) β-
I p,β ≤ R 2 + 1 ∧ 1 x 1 + x 2 p/2 (x 1 x 2 ) β-1-p/2 dx 1 dx 2 ≤ C ∞ 0 x 1 0 1 ∧ 1 x 1 p/2 (x 1 x 2 ) β-1-p/2 dx 1 dx 2 = C ∞ 0 1 ∧ 1 x 1 p/2
x 2β-p- (iii) Follows from stationarity of increments of Z β (part (i)) and J 2,β (τ ) = σ 2 ∞ τ 2 (2-β) , where according to (A.2), (iv) Follows from stationarity of increments, E|Z β (τ )| p ≤ CJ p,β (τ ), 1 < p ≤ 2, where J p,β (τ ) is the same as in (A.3), and Kolmogorov's criterion; c.f ( [START_REF] Pilipauskaitė | Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes[END_REF], proof of Proposition 3.1(iv)).

(v) The proofs are very similar to those of Theorem 3.1 (i), (ii), hence we omit some details. For notational simplicity, we only prove one-dimensional convergence at τ > 0.

Proof of (2. × E[e iθz b (τ ;x 1 ,x 2 ) -1](x 1 x 2 ) β-1 dx 1 dx 2 =:

3 i=1 L i .
By (A.2), we have that E[exp{iθb -1/2 z(bτ ; x 1 , x 2 )} -1] ≤ C min{1, (x 1 x 2 (x 1 + x 2 )) -1 }. In view of (2.8), the dominated convergence theorem applies if the integrands on the r.h.s. of (A.12) converge pointwise, i.e. for every (x 1 , x 2 ) ∈ R 2 + , b -1/2 z(bτ ; x 1 , x 2 ) → d B(τ ) f (s 1 , s 2 ) ≤ Cb -1/2 e -x 1 (s 2 -s 1 ) 1(s 1 < s 2 ) + e -x 2 (s 1 -s 2 ) 1(s 1 ≥ s 2 ) , 0 ≤ s 1 , s 2 ≤ b, (A.

β = 1 '

 1 intermediate Poisson' Arbitrary β > 2 Gaussian a) temporal lags (s = 0) Mutual increase rate of N, n Parameter region Limit distributionN/n 2β → ∞ 1 < β < 3/2 Gaussian 1/2 < β < 1 symmetric (2β)-stable N/n 2β → 0 3/4 < β < 3/2 symmetric (4β/3)-stableN/n 2β → c ∈ (0, -sectional lags (s = 0)

  .37),(3.38) and the dominated convergence theorem imply (3.36) and (3.35).

  be symmetrically handled, this proves (3.43) and (3.30). Proof of (3.22). Since A -1 N,n S † N,n;q (τ ) = Ñq k=1 y kq (τ ) is a sum of Ñq i.i.d. r.v.s y kq (τ ), k = 1, . . . , Ñq , so the first relation in (3.22) follows from Ñq E[e iθy 1 (τ ) -1] → 0, ∀ θ ∈ R. (3.45) Clearly, (3.45) is a direct consequence of (3.29) and the fact that Ñq /N → 0.

  by Rosenthal's inequality, see e.g. ([7], Lemma 2.5.2), and the sum T n = O(n -(p-1) ) = o(1) similarly to (3.57). This proves (3.53), (3.52), and the pointwise convergence in (3.51).

  .61) Relation (3.29) follows from (3.59), (3.61) and the dominated convergence theorem, using the dominating bound

  dy y and use (3.69) together with the fact that |V N,n (θ; y)| ≤ C is bounded uniformly in N, n, y.

  denote the corresponding expectations in (3.67), (3.70). Clearly, (3.69) follows from lim y→∞,y=O(λ) Λ N,n (z; y) = Λ(z), ∀ z > 0, (3.72) and |Λ N,n (z; y)| ≤ C(1 ∧ (1/z)), ∀ 0 < y < λ, 0 < z < y 2 /2. (3.73) The dominating bound in (3.73) is a consequence of (3.64). To show (3.72) use Proposition 2.1 by writing z N,n (τ ; z/y, y ), y := y(1 -z/y 2 ) in (3.67) as the quadratic form: z N,n (τ ; z/y, y

2 by Proposition 2 . 1 .

 21 Conditions on n, y, y , λ in (3.75) are obviously satisfied due to y, y = O(λ) = o(λ ). This proves (3.72) and (3.68), thereby completing the proof of of (3.29).

Theorem 4 . 1 .

 41 Let the mixing distribution satisfy condition (1.2)

Figure 1 :

 1 Figure 1: Density of the limiting random variables in cases [left] (i),(ii), [right] (iv) of Corollary 4.1 for t = 0 and their kernel density estimates constructed from a random sample of size 1000 from γ N,n (0, 0) in (1.6) with N = 5000, a 2 ∼ Beta(2, β), ε(0) ∼ N (0, 1).

Corollary 4 . 2 .

 42 (i) Let the conditions of Theorem 4.1 (i) be satisfied, 1 < β < 2, and βN,n , ψN,n be estimators as in(4.21). Then for any t ∈ Zsup x∈R P N 1-1/ βN,n ( γ N,n (t, 0) -γ(t)) ≤ x -F * βN,n , ψN,n (x) = o p (1). (4.23)(ii) Let the conditions of Theorem 4.1 (ii) be satisfied, 1 < β < 2, and βN,n , ψN,n be estimators as in (4.21).

  x)| = o p (1) follows from (4.21) and continuity of continuity of the c.d.f. F * β,ψ in β, ψ. This proves (4.23). The proof of (4.24), (4.25) is analogous.

2 + 2 ( 0 , 1 ] 2 du 1 du 2 ∞ 0 e 2 = 2 ( 0 , 1 ] 2 |u 1 -

 2201220220121 Ez 2 (1; x 1 , x 2 )dµ β = (ψ(1)/2) -x|u 1 -u 2 | x β-2 dx (ψ(1)/2) 2 Γ(β -1) u 2 | 2(1-β) du 1 du 2 = (ψ(1)/2) 2 Γ(β -1) 2 /((2 -β)(3 -2β)).

2 R 2 + 2 + 2 R 2 +- 1 R 2 + 2 satisfies 1 R 2 + 1 ( 1

 222221221211 [START_REF] Oppenheim | Aggregation of random parameters Ornstein-Uhlenbeck or AR processes: some convergence results[END_REF]. As b → 0, consider Φ b (θ) := log E exp{iθb β-2 Z β (bτ )} = ψ(1)EΨ(θb β-2 z(bτ ; x 1 , x 2 ))(x 1 x 2 ) β-1 dx 1 dx 2 , where Ψ(z) := e iz -1 -iz, z ∈ R. Since b -2 z(bτ ; x 1 , x 2 ) = d z(τ ; bx 1 , bx 2 ), rewrite Φ b (θ) = ψ(1) 2 b -2β R EΨ(θb β z(τ ; x 1 , x 2 ))(x 1 x 2 ) β-1 dx 1 dx 2 , where b -2β Ψ(θb β z(τ ; x 1 , x 2 )) → -(θ 2 /2)z 2 (τ ; x 1 , x 2 ) a.s. Note |b -2β Ψ(θb β z(τ ; x 1 , x 2 ))| ≤ (θ 2 /2)z 2 (τ ; x 1 , x 2 ),where the dominating function satisfies (A.2) and (2.9). Hence, by the dominated convergence theorem,Φ b (θ) → -(θ 2 /2)ψ(1) Ez 2 (τ ; x 1 , x 2 )(x 1 x 2 ) β-1 dx 1 dx 2 = log E{iθσ ∞ B 2-β (τ )}, which finishes the proof. Proof of (2.19) follows that of Thm. 3.1(i), case 0 < β < 1. As b → 0, consider Φ b (θ) := log Ee iθb -1 (log b -1 ) -1/2β Z β (bτ ) = ψ(1) 2 log b E[e iθz b (τ ;x 1 ,x 2 ) -1](x 1 x 2 ) β-1 dx 1 dx 2 , where z b (τ ; x 1 , x 2 ) := b -1 (log b -1 ) -1/(2β) z bτ ; (log b -1 ) -1/(2β) x 1 , (log b -1 ) -1/(2β) x E|z b (τ ; x 1 , x 2 )| 2 ≤ C x 1 x 2 1 ∧ b -1 (log b -1 ) 1/(2β) x 1 + x 2 , (A.7) see (A.2). Split Φ b (θ) = ψ(1) 2 log b -< x 1 + x 2 < b -1 ) + 1(x 1 + x 2 > b -1 ) + 1(x 1 + x 2 < 1)

2x 1 x 2 (x 1 +b 0 f (s 1 , s 2 ) 2 b 0 2 i=1ef 1 f 1 fE[Z 2 kE[Z 2 k 1 (f 2 fff 2 (s 1 , s 2 )ds 1 ds 2 .rewriting R b = d 2 i=1 b 0 b 0 c i (s 1 , s 2 )

 210122211212212212 x 2 ) .(A.[START_REF] Leipus | Orthogonal series density estimation in a disaggregation scheme[END_REF] To simplify notation, let τ = 1 and all b ∈ N. Definez + b (x 1 , x 2 ) := b 0 dB 1 (s 1 )dB 2 (s 2 ), f (s 1 , s 2 ) := b -1/-x i (u-s i ) 1(u > s i )du,andz - b (x 1 , x 2 ) := b -1/2 z(b; x 1 , x 2 ) -z + b (x 1 , x 2 ). Since E(z - b (x 1 , x 2 )) 2 = O(b -1 ) implies z - b (x 1 , x 2 ) = o p (1), we only need to prove thatz + b (x 1 , x 2 ) → d N 0, 1 2x 1 x 2 (x 1 + x 2 ) as b → ∞. (A.14) Write z + b (x 1 , x 2 ) = b k=1Z k as a sum of a sum of a zero-mean square-integrable martingale difference arrayZ k (s 1 , s 2 )dB 1 (s 1 )dB 2 (s 2 ) + (s 1 , s 2 )dB 1 (s 1 )dB 2 (s 2 ) (s 1 , s 2 )dB 1 (s 1 )dB 2 (s 2 ) w.r.t. the filtration F k generated by {B i (s), 0 ≤ s ≤ k, i = 1, 2}, k = 0, . . . , b.By the martingale CLT in Hall and Heyde[START_REF] Hall | Martingale Limit Theory and Its Applications[END_REF], (A.14) then follows fromb k=1 |F k-1 ] → p 1 2x 1 x 2 (x 1 + x 2 ) and b k=1 |Z k | > )] → 0 for any > 0(s 1 , s 2 )ds 1 ds 2 = E(z + b (x 1 , x 2 )) 2 → (2x 1 x 2 (x 1 + x 2 )) -1 , consider R b := b k=1 (E[Z 2 k |F k-1 ] -EZ 2 k ), where E[Z 2 k |F k-1 ] = (s 1 , s 2 )dB 2 (s 2 ) (s 1 , s 2 )dB 1 (s 1 )By dB i (s 1 )dB i (s 2 ) with c 1 (s 1 , s 2 ) = b s 1 ∨s 2 f (s 1 , s)f (s 2 , s)ds, c 2 (s 1 , s 2 ) = b s 1 ∨s 2 f (s, s 1 )f (s, s 2 )ds and using the elementary bound:

0 b 0 c 2 i

 02 16) we obtain E|R b | 2 = 2 i=1 b (s 1 , s 2 )ds 1 ds 2 = O(b -1 ) = o(1), which proves R b = o p (1) and completes the proof of the first relation in (A.15). Finally, using (A.6), (A.16), we obtain b k=1 E|Z k | 4 = O(b -1 ) = o(1), which implies the second relation in (A.15) and completes the proof of (A.14). Proposition 2.2 is proved.

  'cross-sectional' intermediate process. Let dM β ≡ M β (dx 1 , dx 2 , dB 1 , dB 2 ) denote

	Poisson random measure on (R + × C(R)) 2 with mean

  1/3 x i , i = 1, 2. The convergence of the integral in (3.48) follows from (2.8). The explicit value of σ 0 in (3.48) is given in Proposition 2.2 (v) and computed in the Appendix. Recall the representation in(3.31) where B N,n = N 1/(2β) , N/B 2β N,n = 1 and

  .60) see(3.35), where z(τ ; x 1 , x 2 ) is defined in(2.11) and the last expectation in (3.60) is given in (A.2). Then using Skorohod's representation we extend (3.60) to zN,n (τ ; x 1 , x 2 ) → z(τ ; x 1 , x 2 ) a.s. implying also Λ

N,n (τ ; x 1 , x 2 ) → -(θ 2 /2)z 2 (τ ; x 1 , x 2 ) a.s. Since |Λ N,n (θ; τ ; x 1 , x 2 )| ≤ C z2 N,n (τ ; x 1 , x 2 )

and (3.60) holds, by Pratt's lemma we obtain

  .19) see (4.7). Write z * N,n (τ ; x) in (4.15) as quadratic form: z * N,n (τ ; x) = Q 11 (h(τ ; x; •)) in (2.3) and apply Proposition 2.1 with α 1

  1/2 ). By Proposition 2.4 (2.29) we have that var( S t,0 N,n

  Remark 4.4 below. Corollary 4.2 omits the 'intermediate' case λ * ∞ ∈ (0, ∞), partly because in this case the limit distribution is less tractable and depends on λ * ∞ which is difficult to assess in a finite sample.

			)
	the c.d.f.s of the above stable r.v.s, which are uniquely determined by β, ψ(1) ≡ ψ in (1.2), see (4.7)-(4.9).
	The same is true for the (marginal) distribution Z * β (τ ) of the 'diagonal intermediate' process in (2.24). In
	Corollary 4.2 we suppose the existence of estimators
	βN,n = β + o p (1/log N ),	ψN,n = ψ + o p (1),	(4.21)
	σ2 N,n,t = (σ * t ) 2 + o p (1),		(4.22)
	which is discussed in

  1 dx 1 dx 2 , 0 < p ≤ 2, E |z(τ ; x 1 , x 2 )| p ∨ |z(τ ; x 1 , x 2 )| 2 (x 1 x 2 ) β-1 dx 1 dx 2 , p > 2. (A.3) Let first 0 < p ≤ 2. Using E|z(τ ; x 1 , x 2 )| p ≤ (E|z(τ ; x 1 , x 2 )| 2 ) p/2 and (A.2), we obtain J p,β (τ ) ≤ C -(x 1 +x 2 )|u 1 -u 2 | du 1 du 2 p/2 (x 1 x 2 ) β-1-p/2 dx 1 dx 2 =: Cτ 2(p-β) I p,β , (A.4)

		R 2 +
	R 2 +	(0,τ ] 2

e

  3p/4, thus proving (A.3) for 0 < p ≤ 2. Next for 2 < p < 3 we need the inequality for double Itô-Wiener integrals: for anyp ≥ 2, g ∈ L 2 (R 2 ) E R 2 g(s 1 , s 2 )dB 1 (s 1 )dB 2 (s 2 ) p ≤ C E R 2 g(s 1 , s 2 )dB 1 (s 1 )dB 2 (s 2 ) 2 p/2 = C R 2 |g(s 1 , s 2 )| 2 ds 1 ds 2 p/2 . (A.6)Indeed, by using Gaussianity and independence of B 1 , B 2 and Minkowski inequality forI 2 (g) := R 2 g(s 1 , s 2 )dB 1 (s 1 )dB 2 (s 2 ) we obtain E|I 2 (g)| p 2/p = E B 1 E B 2 |I 2 (g)| p B 1 2/p ≤ C E B 1 (E B 2 |I 2 (g)| 2 B 1 ) p/2 2/p ≤ CE B 2 E B 1 |I 2 (g)| p B 2 } 2/p ≤ CE B 2 E B 1 |I 2 (g)| 2 B 2 p/2 2/p = CE B 2 E B 1 |I 2 (g)| 2 B 2 = CE|I 2 (g)| 2 .

		1	1	dx 1 < ∞	(A.5)
	if p/2 < β < Using inequality (A.6) and (A.4), (A.5) we obtain
	J p,β (τ ) ≤ C	R 2

+ E|z(τ ; x 1 , x 2 )| p (x 1 x 2 ) β-1 dx 1 dx 2 + R 2 + E|z(τ ; x 1 , x 2 )| 2 (x 1 x 2 ) β-1 dx 1 dx 2 ≤ C(I p,β (τ ) + I 2,β (τ )) < ∞ if p/2 < β < 3p/4,

thus proving (A.3) and part (ii).

Acknowledgments

The authors are grateful to an anonymous referee and associate editor for useful comments. Vytautė Pilipauskaitė acknowledges the financial support from the project "Ambit fields: probabilistic properties and statistical inference" funded by Villum Fonden.

 (4.23); moreover since the above quantity is non-random, we get that |P(N 1-1/ βN,n ( γ N,n (t, 0) -γ(t)) ≤ q βN,n , ψN,n (α)) -α| = o(1), implying that γ N,n (t, 0) -N (1/ βN,n )-1 q βN,n , ψN,n (1 -α/2), γ N,n (t, 0) -N (1/ βN,n )-1 q βN,n , ψN,n (α/2) is the asymptotic confidence interval for γ(t), for any confidence level α ∈ (0, 1). Analogous confidence intervals for γ(t) can be defined in the case (4.24); in the case (4.25) they follow in a standard way.

Remark 4.4. Estimation of the tail parameter β in the RCAR(1) panel model was studied in [START_REF] Leipus | Estimating long memory in panel randomcoefficient AR(1) data[END_REF]. Particularly, [START_REF] Leipus | Estimating long memory in panel randomcoefficient AR(1) data[END_REF] developed a modified version βN,n of the Goldie-Smith estimator in [START_REF] Goldie | Slow variation with remainder: theory and applications[END_REF] and proved its asymptotic normality, under additional (rather stringent) conditions on the mutual increase rate of N and n. A similar estimator ψN,n can be defined following [START_REF] Goldie | Slow variation with remainder: theory and applications[END_REF]. We expect that these estimators satisfy the consistency as in (4.21) under much weaker assumptions on N, n. Finally, for t ≥ 0 the estimator σ2 N,n,t in (4.22) can be defined (see the proof in Appendix) as

Remark 4.5. In general, in the RCAR(1) model the autoregressive coefficient a can take a value from (-1, 1).

In the latter case if the distribution of a is sufficiently dense at -1, the (unconditional) autocovariance function of the RCAR(1) process oscillates when decaying slowly, which is usually referred to as seasonal long memory. The restriction a ∈ [0, 1) in the present paper (as well as in [START_REF] Pilipauskaitė | Scaling transition for nonlinear random fields with long-range dependence[END_REF], [START_REF] Leipus | Estimating long memory in panel randomcoefficient AR(1) data[END_REF] and some other papers) is basically due to technical reasons. We expect that, under assumption (1.2), most of our results hold in the general case a ∈ (-1, 1) provided the concentration of the mixing distribution near -1 is not too strong, e.g., if E(1 + a) -β < ∞ for some β > β.

A Appendix

Proof of Proposition 2.2. (i) The existence of Z β follows from

and

Using (A.7), we can show that L i , i = 2, 3 are remainders. By change of variables: y = x 1 + x 2 , x 1 = yw and then w = z/y 2 , we rewrite the main term

Here the dominating bound is a consequence of (A.7). Then

where Λ(z

for more details we refer the reader to the proof of Thm. 3.1 (i) case 0 < β < 1. More precisely, (A.10) says that for every > 0 there exists a small δ > 0 such that for all 0

We have that z b (τ ; z y , y(1

where

Conditions on b, y, y are obviously satisfied due to y, y = O(b -1 ) = o(b -1 (log b -1 ) 1/(2β) ). This proves (A.10) and (A.9), thereby completing the proof of of (2.19).

Proof of (2.20) follows that of Theorem 3.1 (ii). We will prove that as b → ∞,

where

where 0 < p ≤ 1 for β ∈ (0, 1) and 1 ≤ p ≤ 2 for β ∈ (1, 2). We have E|z * (τ ; x) -τ /2x| p ≤ (var(z * (τ ; x))) p/2 , where [START_REF] Mikosch | Is network traffic approximated by stable Lévy motion or fractional Brownian motion?[END_REF], where p is sufficiently close to β and such that 0 < p < β < 3p/2. This proves part (ii).

This completes the proof of part (i). (ii) E|V

(iii) Follows from part (ii) by Kolmogorov's criterion, similarly as in the proof of Proposition 2.2.

(iv) For notational simplicity, we only prove one-dimensional convergence at τ > 0.

We have log

we obtain the bounds:

The result then follows from the dominated convergence theorem once we show that for all x ∈ R + ,

where Z ∼ N (0, 1). Using (A.18), we get E|b Calculation of the constant σ 0 in Proposition 2.2 (v). We have

Proof of (4.22). By Corollary 4.1 (iv), 1

(A.20)

By the LLN, and the same bound as in (2.29) we see that the l.h.s. of (A.21) does not exceed CE[ 1 (1-a i ) 2 min{1, 1 n(1-a i ) }] which vanishes as n → ∞ by the dominated convergence theorem, due to E(1 -a) -2 < ∞.