Joint Asymptotics for Estimating the Fractal Indices of Bivariate ... - arXiv

arXiv:1609.03470v1 [math.ST] 12 Sep 2016

Joint Asymptotics for Estimating the Fractal Indices of Bivariate Gaussian Processes∗ Yuzhen Zhou1 and Yimin Xiao2 1 University

of Nebraska-Lincoln e-mail: [email protected]

2 Michigan

State University e-mail: [email protected]

Abstract: For modeling multiple variables, the fractal indices of the components of the underlying multivariate process play a key role in characterizing the dependence structure among the components and statistical properties of the multivariate process. In this paper, under the infill asymptotics framework, we establish joint asymptotic results for the increment-based estimators for bivariate fractal indices. Our main results show the effect of the cross dependence structure on the performance of the estimators. Keywords and phrases: Fractal Indices, Bivariate Gaussian Process, Bivariate Mat´ ern Field, Joint Asymptotics.

1. Introduction The fractal index of a stochastic process is very useful for measuring the roughness of its sample paths (e.g., it determines the Hausdorff dimension of the trajectories of the process) and is an important parameter for geostatistics modeling. The problem on estimating the fractal index of a real-valued Gaussian and non-Gaussian process has attracted the attention of many authors in the last decades. Hall and Wood (1993) studied the asymptotic properties of the boxcounting estimator for the fractal index. Constantine and Hall (1994) introduced the variogram method for studying the problem. Kent and Wood (1997) developed the increment-based estimators for stationary Gaussian process, which can achieve improved performance under infill asymptotics (namely, asymptotic properties of statistical procedures as the sampling points grow dense in a fixed domain, see Chen, Simpson and Ying (2000); Cressie (1993)). Chan and Wood (2000, 2004) extended the method to a class of stationary Gaussian random fields on R2 and their transformations which are non-Gaussian in general. Zhu and Stein (2002) expanded the work of Chan and Wood (2000) by considering the fractional Brownian surface. More recently, Coeurjolly (2008) introduced a new class of consistent estimators of the fractal dimension of locally self-similar Gaussian processes on R using sample quantiles and derived the almost sure convergence and the asymptotic normality for these estimators. Bardet et al. (2011) provided estimators for the fractal index based on increment ratios for ∗ Research

supported in part by NSF grants DMS-1612885 and DMS-1607089. 1

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several classes of real-valued processes with rough sample paths including Gaussian processes and studied their asymptotic properties. Loh (2015) constructed estimators from irregularly spaced data on Rd with d = 1 or 2 via higher-order ˇ c´ıkov´a and Percival (2012) and quadratic variations. We refer to Gneiting, Sevˇ the references therein for further information on various types of estimators and their assessments. In recent years, multivariate Gaussian processes and random fields have been popular in modeling multivariate spatial datasets (see, e.g., Gelfand et al. (2010), Wackernagel (2003)). Several classes of multivariate spatial models have been introduced by Gneiting, Kleiber and Schlather (2010), Apanasovich, Genton and Sun (2012) and Kleiber and Nychka (2012). Two of the challenges in multivariate modeling are to specify the cross-dependence structures and to quantify the effect of the cross-dependence on the performance of estimation and prediction. We refer to Genton and Kleiber (2015) for an excellent review on recent development in multivariate covariance functions. They also raised many open questions and called for theoretical development of estimation and prediction methodology in the multivariate context. So far only a few authors have worked in this direction; see, for example, Pascual and Zhang (2006), Lim and Stein (2008), and Zhang and Cai (2015). While focusing on the fractal dimensions of multivariate Gaussian process, we’ve only seen one piece of work by Amblard and Coeurjolly (2011) who constructed estimators of fractal indices for multivariate fractional Brownian motions by using the discrete filtering techniques and studied their joint asymptotic distribution. By estimating fractal index of each component separately, they found the quality of these estimations was almost independent of the cross correlation of the multivariate fractional Brownian motion. In this work, we consider a class of bivariate stationary Gaussian processes X , {(X1 (t), X2 (t))> , t ∈ R} and study the joint asymptotic properties of the estimators for the fractal indices of the components X1 and X2 under infill asymptotics framework. Our main purpose is to clarify the effect of the cross covariance on the performance of the joint estimators. More specifically, we assume that X has mean EX(t) = 0 and matrix-valued covariance function   C11 (t) C12 (t) C(t) = , (1.1) C21 (t) C22 (t) where Cij (t) := E[Xi (s)Xj (s + t)], i = 1, 2. Further, we assume that the following conditions are satisfied C11 (t) = σ12 − c11 |t|α11 + o(|t|α11 ), C22 (t) = σ22 − c22 |t|α22 + o(|t|α22 ), C12 (t) = C21 (t) = ρσ1 σ2 (1 − c12 |t|

(1.2) α12

α12

+ o(|t|

)),

where α11 , α22 ∈ (0, 2), σ1 , σ2 > 0, |ρ| ∈ (0, 1) and c11 , c22 , c12 > 0 are constants. Under assumption (1.2), in order for (1.1) to be a valid covariance function, it is necessary to impose some restrictions on the parameters (α11 , α22 , α12 ). In this

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paper, we assume α11 + α22 < α12 , or 2 α11 + α22 = α12 and c212 ρ2 σ12 σ22 < c11 c22 . 2

(1.3)

This is a very mild assumption. See (A) in Appendix for a justification. From now on, we name (1.2) and (1.3) as Condition (A1). Under assumption (1.2), it is well known (see, e.g., Adler (1981), Theorem 8.4.1) that the fractal dimensions of the trajectories of each component X1 and X2 are given by dim GrX1 ([0, 1]) = 2 −

α11 , 2

a.s.

and

α22 , a.s., 2 respectively. In the above, for i ∈ {1, 2}, GrXi ([0, 1]) = {(t, Xi (t)) : t ∈ [0, 1]} is the trajectory (or graph set) of the real-valued process Xi = {Xi (t), t ∈ R} over the interval [0, 1]. A bivariate stationary Gaussian process X = {(X1 (t), X2 (t))> , t ∈ R} with matrix-valued covariance function (1.1) that satisfies Condition (A1) has richer fractal properties. For example, we consider the trajectory of X on [0, 1] which is GrX([0, 1]) = {(t, X1 (t), X2 (t))> : t ∈ [0, 1]}. For notational convenience, we further assume that α11 ≤ α22 (otherwise we may relabel the components of X). Then we can apply Theorem 2.1 in Xiao (1995) to show that with probability 1,   2 + α22 − α11 α11 + α22 dim GrX([0, 1]) = min , 3− α22 2  2+α22 −α11 (1.4) , if α11 + α22 ≥ 2, α 22 = 22 3 − α11 +α , if α11 + α22 < 2. 2 dim GrX2 ([0, 1]) = 2 −

This result shows that the indices α11 and α22 determine the fractal dimension of the trajectory of the bivariate Gaussian process X. Furthermore, one can characterize many other fractal properties of X explicitly in terms of these indices. See Xiao (2013) for a recent overview. Hence, analogous to the univariate case, it is natural to call (α11 , α22 ) the fractal indices of X. For reader’s convenience, we include a proof of (1.4) in (B) of the Appendix. Even though the parameters α11 and α22 can be estimated separately from observations of the coordinate processes X1 and X2 , (1.4) suggests that, in doing so, one might miss some important information on the structures of the bivariate process X. For example, although the estimator of dim GrX([0, 1]) can be obtained by plugging in the estimators of α11 and α22 to (1.4), say (α ˆ 11 , α ˆ 22 ), we cannot evaluate the estimation efficiency without the joint asymptotic properties of (α ˆ 11 , α ˆ 22 ). Hence it is necessary to study these estimators jointly and to quantify the effect of the cross-covariance on their performance. In this paper, we will work with the increment-based estimators of α11 and α22 , denoted by α ˆ 11 and α ˆ 22 , respectively, and study the bias, mean square

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error matrix and joint asymptotic distribution under the infill asymptotics framework. The main√ results are√shown in Theorems 3.3 ∼ 3.5. In particα11 and nˆ α22 are asymptotically uncorrelated for ular, we prove that nˆ the case when 12 (α11 + α22 ) < α12 , while they are asymptotically correlated if 12 (α11 + α22 ) = α12 . Our results are applicable to a wide class of bivariate Gaussian processes including those with Mat´ern cross-covariances introduced by Gneiting, Kleiber and Schlather (2010). Further, the method on joint asymptotics developed in this paper and the recent work by Loh (2015) on constructing estimators for univariate fractal index given irregularly spaced data make it possible to study the joint asymptotics of estmating bivariate fractal indices when data were observed irregularly on R2 , which would be a challenging and interesting problem. The rest of the paper is organized as follows. We follow Kent and Wood (1997), Chan and Wood (2000) and formulate the increment-based estimators for (α11 , α22 ) in Section 2. In Section 3, we state the main results on the joint asymptotics of the bivariate estimators. An application to the nonsmooth bivariate Mat´ern processes is shown in Section 4. In Section 5, we offer a simulation study on the efficiency of the estimators. The proofs of our main results are given in Section 6. Finally, some auxiliary results and their proofs are included in the Appendix. We end the introduction with some notation. Z+ denotes the set of all positive integers and B(R) is the collection of all Borel sets on R. For any real-valued ∞ sequences {an }∞ n=1 , {bn }n=1 , an ∼ bn means limn→∞ bn /an = 1, an & bn means that there exists a constant c > 0 such that an ≥ c bn for all n large enough and an  bn means an & bn and bn & an . Similar notation is also used for functions of continuous variables. An unspecified positive and finite constant will be denoted by C0 . More specific constants are numbered by C1 , C2 , . . . . 2. The increment-based estimators Assume that the values of bivariate process X = {(X1 (t), X2 (t))> , t ∈ R} are observed regularly on an interval I, say I = [0, 1]. Specifically, we have n pairs of observations (X1 ( n1 ), X2 ( n1 )), (X1 ( n2 ), X2 ( n2 )), ..., (X1 (1), X2 (1)). By applying the increment-based method introduced by Kent and Wood (1997) for estimating the fractal index of a real-valued locally self-similar Gaussian process (see also Chan and Wood (2000, 2004) for further development), we can estimate the fractal indices (α11 , α22 ) of X. Our emphasis is on studying the joint asymptotic properties of the estimators. In particular, we study the effect of cross-covariance on their joint performance. For each component Xi , i = 1, 2 and an integer u ∈ {1, 2, ..., m} with m given, we define the dilated filtered discretized process with second difference (see, e.g., Kent and Wood (1997)),   j   j + u  αii j − u u 2 Xi − 2Xi + Xi , j = 1, ..., n. (2.1) Yn,i (j) := n n n n

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u Denote by a−1 = 1, a0 = −2, a1 = 1. Yn,i can be rewritten as u Yn,i (j)

=n

αii 2

1 X

 ak Xi

k=−1

 j + ku . n

(2.2)

u As in Kent and Wood (1997), one can verify that, under (1.2), Yn,i (j) is a Gaussian random variable with mean 0 and its variance converges to cii (8 − u u 2αii +1 )uαii (this follows from (3.4) below). Let Zn,i (j) := (Yn,i (j))2 and define n

1X u u Z (j). Z¯n,i := n j=1 n,i

(2.3)

For i = 1, 2, it follows from Kent and Wood (1997) that, under certain regularity conditions on the covariance function Cii (t), we have u p Z¯n,i − → Ci uαii , p

where − → means convergence in probability and Ci = cii (8 − 2αii +1 ). Hence u p log Z¯n,i − → αii log u + log Ci , i = 1, 2.

(2.4)

Hence the fractal indices αii (i = 1, 2) can be estimated by linear regression of u log Z¯n,i on log u for u = 1, 2, . . . , m. In this paper, we employ Chan and Wood (2000)’s linear estimators for αii based on log Z¯ u , that is n,i

α ˆ ii =

m X

u Lu,i log Z¯n,i ,

(2.5)

u=1

where {Lu,i , u = 1, 2, ..., m} (i = 1, 2) are finite sequences of real numbers such that m m X X Lu,i log u = 1. (2.6) Lu,i = 0 and u=1

u=1

Both the ordinary least square and the generalized least square estimators introduced by Kent and Wood (1997) are examples of the above estimators. 3. Joint asymptotic properties For i = 1, 2, let 1 2 m > ¯ n,i = (Z¯n,i Z , Z¯n,i , ..., Z¯n,i )

(3.1)

¯ n = (Z ¯> ¯> > Z n,1 , Zn,2 ) .

(3.2)

and denote

Under the infill asymptotics framework, we first study the asymptotic properties ¯ n in Section 3.1. In Section 3.2, the joint asymptotic properties of the of Z estimators (α ˆ 11 , α ˆ 22 )> are obtained.

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¯ n and asymptotic normality 3.1. Variance of Z u v > First, given u, v = 1, 2, ..., m, we consider the covariance matrix of (Yn,1 , Yn,2 ) . For i = 1, 2, it follows from Kent and Wood (1997) that the marginal covariance u v function for Yn,i and Yn,i is uv u v σn,ii (h) := E[Yn,i (l)Yn,i (l + h)]

→ −cii

1 X

uv aj ak |h + kv − ju|αii , σ0,ii (h),

(3.3)

j,k=−1 u as n → ∞. In particular, we derive that the variance of Yn,i (l) satisfies uu σn,ii (0) → Ci uαii ,

as n → ∞,

(3.4)

where Ci = cii (8 − 2αii +1 ). u v Under Assumption (1.2), the cross covariance between Yn,1 and Yn,2 can be derived as follows. uv u v σn,12 (h) := E[Yn,1 (l)Yn,2 (l + h)]   1 X α11 +α22 h + kv − ju 2 aj ak C12 =n n j,k=−1  0, if P1 → −ρσ1 σ2 c12 j,k=−1 aj ak |h + kv − ju|α12 , if

α11 +α22 2 α11 +α22 2

(3.5) < α12 , = α12

uv , σ0,12 (h). u v Therefore, if (α11 +α22 )/2 < α12 , the covariance matrix of (Yn,1 (l), Yn,2 (l +h))> satisfies  u   uu  Yn,1 (l) σ0,11 (0) 0 Var → , as n → ∞. (3.6) v vv Yn,2 (l + h) 0 σ0,22 (0) u v If (α11 + α22 )/2 = α12 , the covariance matrix of (Yn,1 (l), Yn,2 (l + h))> satisfies  u   uu  uv Yn,1 (l) σ0,11 (0) σ0,12 (h) Var → , as n → ∞. (3.7) v uv vv Yn,2 (l + h) σ0,12 (h) σ0,22 (0)

We adapt the method of derivation in Section 3 of Kent and Wood (1997) to ¯ n . Using the fact that, if find the covariance matrix of the random vector Z (U, V ) ∼ Normal(0, 0, 1, 1, ξ), then Cov(U 2 , V 2 ) = 2ξ 2 , we obtain u v uv Cov(Zn,i (l), Zn,j (l + h)) = 2(σn,ij (h))2 ,

i, j = 1, 2,

and hence φuv n,ij

1 u v := Cov(Z¯n,i , Z¯n,j )= n

n−1 X h=−n+1



|h| 1− n



uv × 2(σn,ij (h))2 .

(3.8)

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m ¯ ¯ Denote by Φn,ij = (φuv n,ij )u,v=1 the covariance matrix of Zn,i and Zn,j . Then, ¯ the covariance matrix of Zn can be written as   Φn,11 Φn,12 Φn = . (3.9) Φn,21 Φn,22

¯ n , we impose an adIn order to study the asymptotic properties of Φn and Z ditional regularity condition on the fourth derivative of the functions Cij (t) in (1.1), which will be called Condition (A2): c11 α11 ! |t|α11 −4 + o(|t|α11 −4 ), (α11 − 4)! c22 α22 ! (4) |t|α22 −4 + o(|t|α22 −4 ), C22 (t) = − (α22 − 4)! c12 α12 ! (4) (4) C12 (t) = C21 (t) = −ρσ1 σ2 |t|α12 −4 + o(|t|α12 −4 ). (α12 − 4)! (4)

C11 (t) = −

α! In the above, for any α > 0, (α−4)! = α(α − 1)(α − 2)(α − 3). P∞ uv uv For i, j = 1, 2, let φ0,ij = 2 h=−∞ (σ0,ij (h))2 which is convergent, Φ0,ij = uv m (φ0,ij )u,v=1 , and let   Φ0,11 Φ0,12 Φ0 = . (3.10) Φ0,21 Φ0,22

The following theorems describe the asymptotic properties of the random ¯n. vector Z Theorem 3.1. If Conditions (A1) and (A2) hold, then nΦn → Φ0 , as n → ∞.

(3.11)

Moreover, if (α11 + α22 )/2 < α12 , then Φ0,12 = Φ0,21 = 0. Theorem 3.2. If Conditions (A1) and (A2) hold, then d ¯ n − E[Z ¯ n ]) − n1/2 (Z → N2m (0, Φ0 ), as n → ∞,

(3.12)

where N2m (0, Φ0 ) is the (2m)-dimensional normal distribution with mean 0 and covariance matrix Φ0 . Remark 3.1. The above theorems extend 2 in Kent and Wood (1997) √ Theorem ¯ n,1 and √nZ ¯ n,2 are asymptotically to the bivariate case, which show that nZ independent when (α11 + α22 )/2 < α12 . The proofs of Theorems 3.1 and 3.2 will be given in Appendix D. 3.2. Asymptotic properties of (α ˆ 11 , α ˆ 22 )> This section contains the main results of this paper. We will make a stronger assumption by specifying the remainder terms in Assumption (1.2). Suppose

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that, for some constants β11 , β22 , β12 > 0, C11 (t) = σ12 − c11 |t|α11 + O(|t|α11 +β11 ), C22 (t) = σ22 − c22 |t|α22 + O(|t|α22 +β22 ), α12

C12 (t) = C21 (t) = ρσ1 σ2 (1 − c12 |t|

(3.13) α12 +β12

+ O(|t|

)).

We label the three conditions in (3.13), together with (1.3), as Condition (A3). ˆ = (ˆ Let α α11 , α ˆ 22 )> be the estimators of the fractal indices α = (α11 , α22 )> , as defined in (2.5). The theorems below establish asymptotic properties of (α ˆ 11 , α ˆ 22 )> , including the bias, mean square error matrix and their joint asymptotic distribution. Theorem 3.3 (Bias). Assume Conditions (A2) and (A3) hold. Then for the estimators α ˆ ii , i = 1, 2, we have   E α ˆ ii − αii = O(n−1 ) + O(n−βii ), i = 1, 2. (3.14) Theorem 3.4 (Mean square error matrix). Assume (A2) and (A3) hold. If (α11 + α22 )/2 = α12 , then E[(α ˆ − α)(α ˆ − α)> ]     O(n−1 )) O(n−1 ) O(n−ψ(β11 ,β11 ) ) O(n−ψ(β11 ,β22 ) ) = + . (3.15) O(n−1 ) O(n−1 ) O(n−ψ(β11 ,β22 ) ) O(n−ψ(β22 ,β22 ) ) Here and below, ψ(x1 , x2 ) := min{1 + x1 , 1 + x2 , x1 + x2 }. If (α11 + α22 )/2 < α12 , then ˆ − α)(α ˆ − α)> ] E[(α     O(n−1 ) o(n−1 ) O(n−ψ(β11 ,β11 ) ) O(n−ψ(β11 ,β22 ) ) = + . o(n−1 ) O(n−1 ) O(n−ψ(β11 ,β22 ) ) O(n−ψ(β22 ,β22 ) )

(3.16)

Remark 3.2. The constants β11 , β22 and β12 from (3.13) appear in both bias and mean square error matrix (Theorems 3.3 and 3.4) because of the remainder terms O(|t|αii +βii ) in the covariance function are ignored in the procedure of estimation, which might heavily affect the efficiency of the estimators (see, e.g., Kent and Wood (1997)). The statistical performance of the estimators (ˆ α11 , α ˆ 22 )> can be significantly improved if more detailed information on the remainder term is available. In Section 4, we will show that this is indeed the case when X = {(X1 (s), X2 (s))> , s ∈ R} is a nonsmooth bivariate Mat´ern process. ˆ by applying mulFinally, we study the asymptotic distribution of the α tivariate delta method (see, e.g., Lehmann and Casella (2006); Amblard and Coeurjolly (2011)). By (2.3), (3.3), and (3.13), we have u u EZ¯n,i = E[(Yn,i (0))2 ] = τu,i (1 + O(n−βii )),

(3.17)

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where τu,i = cii (8 − 2αii +1 )uαii . Let e i = (L1,i /τ1,i , L2,i /τ2,i , ..., Lm,i /τm,i )> , L

i = 1, 2.

ˆ The following theorem provides the joint asymptotic distribution of α. Theorem 3.5 (Asymptotic distribution). Assume (A2) and (A3) hold with √ ˆ − α) follows the asymptotic properties below. β11 , β22 > 21 . Then, n(α !!     e > Φ0,11 L e1 L e > Φ0,12 L e2 √ α ˆ 11 − α11 0 L d 1 1 n − →N , . (3.18) e > Φ0,21 L e1 L e > Φ0,22 L e2 α ˆ 22 − α22 0 L 2 2 Especially, if pendent.

α11 +α22 2

< α12 , then

√

nˆ α11 and

√

nˆ α22 are asymptotically inde-

4. An example: nonsmooth bivariate Mat´ ern processes on R The Mat´ern correlation function M (h|ν, a) on RN , where a > 0, ν > 0 are scale and smoothness parameters, is widely used for modeling covariance structures in spatial statistics. It is defined as M (h|ν, a) :=

21−ν (a|h|)ν Kν (a|h|), Γ(ν)

h ∈ RN ,

(4.1)

where Kν is a modified Bessel function of the second kind. Recently Gneiting, Kleiber and Schlather (2010) have introduced the full bivariate Mat´ern fields X = {(X1 (s), X2 (s))> , s ∈ RN }, which is an R2 -valued Gaussian random field on RN with zero mean and matrix-valued covariance functions:   C11 (h) C12 (h) C(h) = , (4.2) C21 (h) C22 (h) where Cij (h) := E[Xi (s + h)Xj (s)] are specified by C11 (h) = σ12 M (h|ν11 , a11 ), C22 (h) = σ22 M (h|ν22 , a22 ),

(4.3)

C12 (h) = C21 (h) = ρσ1 σ2 M (h|ν12 , a12 ). A necessary and sufficient condition for C(h) in (4.2) to be valid is given by Gneiting, Kleiber and Schlather (2010). We assume that the parameters νij , aij , σi , (i, j = 1, 2) and ρ satisfy the condition in Theorem 3 of Gneiting, Kleiber and Schlather (2010) as well as our condition (1.3). To apply the results in Section 3, we focus on the case of N = 1 and 0 < ν11 , ν22 < 1. Then X = {(X1 (s), X2 (s))> , s ∈ R} is a stationary bivariate Gaussian process with nonsmooth sample functions. For simplicity we will call X a bivariate Mat´ern process.

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Recall that the Mat´ern correlation function has the following asymptotic expansion at h = 0, M (h|ν, a) = 1 − b1 |h|2ν + b2 |h|2 + O(|h|2+2ν ),

as |h| → 0,

(4.4)

where b1 , b2 are explicit constants depending on ν and a only (Eq. (4.4) follows from (9.6.2) and (9.6.10) in Abramowitz and Stegun (1972)). So Assumption (A3) is satisfied with βij = 2 − νij for i, j = 1, 2. Moreover, one can check that the regularity condition (A2) regarding the fourth derivatives of the covariance function is also satisfied (see the proof in Appendix C). P1 Thanks to (4.4) and the fact that j,k=−1 aj ak |k − j|2 = 0, we have uu σn,ii (0)

:=

u E[(Yn,i (0))2 ]

=n

2νii

1 X

 aj ak Cii

j,k=−1 2 = −b1 σii

1 X

(k − j)u n



aj ak |k − j|2νii u2νii + O(n−2 ).

(4.5)

j,k=−1

Observe that, unlike (3.17), the constants βij = 2 − νij do not appear in (4.5) because the related terms sum to 0. Consequently we can prove the following results that are stronger than what can be obtained by applying directly Theorems 3.3 ∼ 3.5 to bivariate Mat´ern processes. Their proofs are modifications of those of Theorems 3.3 ∼ 3.5 in Section 6 and will be omitted. Proposition 4.1 (Bias). Let X = {(X1 (s), X2 (s))> , s ∈ R} be a bivariate Mat´ern process with 0 < ν11 , ν22 < 1. The bias of νˆii is   E νˆii − νii = O(n−1 ), i = 1, 2. (4.6) ˆ = (ˆ For the next proposition, we write ν = (ν11 , ν22 )> and ν ν11 , νˆ22 )> . Proposition 4.2 (Mean square error matrix). Let X = {(X1 (s), X2 (s))> , s ∈ R} be a bivariate Mat´ern process with 0 < ν11 , ν22 < 1. If (ν11 + ν22 )/2 = ν12 , then   O(n−1 ) O(n−1 ) > E(ˆ ν − ν)(ˆ ν − ν) = . (4.7) O(n−1 ) O(n−1 ) If (ν11 + ν22 )/2 < ν12 , we have >

E(ˆ ν − ν)(ˆ ν − ν) =



O(n−1 ) o(n−1 ) o(n−1 ) O(n−1 )

 .

(4.8)

Proposition 4.3 (Asymptotic distribution). For the nonsmooth bivariate Mat´ern process with 0 < ν11 , ν22 < 1, !!     e > Φ0,11 L e1 L e > Φ0,12 L e2 √ νˆ11 − ν11 0 L d 1 1 n − →N . (4.9) , e > Φ0,21 L e1 L e > Φ0,22 L e2 νˆ22 − ν22 0 L 2 2 √ √ 22 Especially, if ν11 +ν < ν12 , nˆ ν11 and nˆ ν22 are asymptotically independent. 2

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5. Simulation Study In this section, we simulate the data from a nonsmooth bivariate Mat´ern process 22 and first illustrate that when ν11 +ν = ν12 , the decay rates of bias and mean 2 square error matrix for νˆ11 and νˆ22 are n−1 . Then we compare it with the case 22 < ν12 . when ν11 +ν 2 We take ν11 = 0.2, ν22 = 0.7, ν12 = 0.45, ρ = 0.5, σ12 = σ22 = 1 and a11 = a22 = a12 = 1. We have simulated the corresponding bivariate Mat`ern process on regular grids within the interval [0, 1], where the length of the grid is set to be 1/n with n = 200, 210, 220, ..., 1000. For each n, we used the generalized least square (abbr. GLS ) method to obtain the estimators of fractal indices, say (ˆ ν11 , νˆ22 ) (see, e.g., Kent and Wood (1997)). Here, we fixed the number of dilations m = 50. The weight matrix Ωi = (ωiuv )m u,v=1 of the GLS estimator with Mat´ern covariance function is given by Pn−u Pn−v P1 2νii 2 ) 2 h=u l=v ( j,k=−1 aj ak |h − l + kv − ju| uv , ωi = P1 2ν 2 2ν 2ν (n − 2u + 1)(n − 2v + 1) ( j,k=−1 aj ak |k − j| ii ) u ii v ii which can be approximated by plugging the ordinary least square estimators of νii , i = 1, 2. To evaluate the efficiency of the estimators, we repeated the above procedure 1000 times independently. The 95% confidence intervals for (ν11 , ν22 ) with varying n are shown in FIG 1 (a). FIG 1 (c) and (e) show how the bias, marginal variances and cross covariance decrease when n increases from 200 to 1000. Fitting the log of absolute value of bias, marginal variance and absolute values of cross covariance with respect to log n respectively, we find the power of decay rate for each of them is very close to −1. This is consistent with the conclusions in Proposition 4.1 and Proposition 22 = ν12 . 4.2 when ν11 +ν 2 22 Further, we show how the decay rates changes if ν11 +ν < ν12 . Fixing all 2 the other parameters assigned previously but setting ν12 to be 0.6, we rerun the simulation and repeated the estimation procedures. The results are shown on the right hand side of FIG 1, from where we can see everything stays almost the same but the cross covariance decays much faster than n−1 . Indeed, we have found the power of decay rate is about −1.5, which is consistent with the conclusion in Proposition 4.2. 6. Proof of the main results For proving Theorems 3.3 ∼ 3.5, we will make use of the following key lemma. u Lemma 6.1. For u = 1, 2, ..., m and i = 1, 2, let Tn,i =

¯ u −EZ ¯u Z n,i n,i . ¯u EZ n,i

Then, for

+

any k ∈ Z , there exist positive and finite constants C3 , C4 (which may depend on u and k) such that for all n ≥ 1 and ξ > 0, √  u k u E log(1 + Tn,i ) ; |Tn,i | > ξ] ≤ C3 e−C4 ξ n .

(6.1)

1.0

1.0

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

0.2

0.4

0.6

0.8

95% CI for nu11 95% CI for nu22 true nu

0.0

0.0

0.2

0.4

0.6

0.8

95% CI for nu11 95% CI for nu22 true nu

12

200

400

600

800

1000

200

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0.004

0.006

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0.012

bias for nu11 bias for nu22

0.002

0.002

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0.006

0.008

0.010

0.012

bias for nu11 bias for nu22

0.014

(b)

0.014

(a)

600

200

400

600

800

1000

200

400

(c)

800

1000

(d)

0.0020 0.0015 0.0010 0.0005 0.0000

0.0000

0.0005

0.0010

0.0015

0.0020

marginal variance for nu11 marginal variance for nu22 cross covariance

0.0025

marginal variance for nu11 marginal variance for nu22 cross covariance

0.0025

600

200

400

600

(e)

800

1000

200

400

600

800

1000

(f)

Fig 1. Confidence intervals, absolute value of bias, marginal variances and absolute value of cross covariance for (ˆ ν11 , νˆ22 ) with varying n. Plots on the left hand side (i.e., a, c, e) show 22 the results for the case when ν11 +ν = ν12 , while those on the right hand side (i.e., b, d, f ) 2 22 correspond to the situation when ν11 +ν < ν12 . 2

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

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The proof of Lemma 6.1 will be given at the end of this section. Now we proceed to prove our main theorems. u u u u Proof of Theorem 3.3. Recall that Tn,i = (Z¯n,i − EZ¯n,i )/EZ¯n,i . Then

α ˆ ii =

m X

u Lu,i log(1 + Tn,i )+

u=1

m X

u Lu,i log EZ¯n,i .

(6.2)

u=1

It follows from (3.17) that u uu EZ¯n,i = σn,ii (0) = Ci uαii .(1 + O(n−βii )),

Hence, by using the conditions on Lu,i in (2.6), we conclude m X

u Lu,i log EZ¯n,i = αii + O(n−βii ).

(6.3)

u=1

Next, we estimate the first sum in (6.2). By Taylor expansion, we have 1 u 2 u u u log(1 + Tn,i ) = Tn,i − (Tn,i ) + Rn,i , 2 u where Rn,i is the residual term. Hence,

1 u 2 1 u u u u 2 u E[log(1 + Tn,i )] = E[Tn,i − (Tn,i ) + Rn,i ] = − E(Tn,i ) + ERn,i . 2 2

(6.4)

By Theorem 3.1, it is easy to check that u 2 E(Tn,i ) = O(n−1 ).

(6.5)

u u u 2 Using the fact that if |Tn,i | ≤ ξ ≤ 21 , then |Rn,i | ≤ ξ(Tn,i ) , we have u u u 2 u E[|Rn,i |; |Tn,i | ≤ ξ] ≤ ξE[(Tn,i ) ; |Tn,i | ≤ ξ] = O(n−1 ),

(6.6)

where the last equality follows from (6.5). On the other hand, applying Lemma 6.1, the Cauchy-Schwarz inequality and (6.5), we obtain u u E[|Rn,i |; |Tn,i | > ξ]

1 u 2 u u u ≤ E[| log(1 + Tn,i )| + |Tn,i | + (Tn,i ) ; |Tn,i | > ξ] = O(n−1 ). 2

(6.7)

Combining (6.5), (6.6) and (6.7), we have u E[log(1 + Tn,i )] = O(n−1 ).

This, together with (6.2) and (6.3), proves Theorem 3.3.

(6.8)

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

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  Proof of Theorem 3.4. For i = 1, 2, we expand E (α ˆ ii − αii )2 as follows. m X m h   X
2 u u E (ˆ αii − αii ) = Lu,i Lv,i E log(1 + Tn,i ) + log EZ¯n,i − αii log u u=1 v=1 v v × log(1 + Tn,i ) + log EZ¯n,i − αii log v

=

m X m X

  u v Lu,i Lv,i E log(1 + Tn,i ) log(1 + Tn,i )

u=1 v=1 m X m X

+ + +


i

u=1 v=1 m X m X u=1 v=1 m m X X

 
u v Lu,i Lv,i E log(1 + Tn,i ) log EZ¯n,i − αii log v
  u v Lu,i Lv,i log EZ¯n,i − αii log u E log(1 + Tn,i )

u v Lu,i Lv,i log EZ¯n,i − αii log u log EZ¯n,i − αii log v

u=1 v=1

, I + II + III + IV.

(6.9)

By (6.3) and (6.8), we have II = O(n−1−βii ), III = O(n−1−βii ), IV = O(n−2βii ).

(6.10)

To bound the first term I in (6.9), we take ξ = 1/2 and decompose the probability u v space into the union of the following four disjoint events, i.e., {|Tn,i | ≤ ξ, |Tn,i |≤ u v u v u v ξ}, {|Tn,i | > ξ, |Tn,i | ≤ ξ}, {|Tn,i | ≤ ξ, |Tn,i | > ξ}, and {|Tn,i | > ξ, |Tn,i | > ξ}, respectively. u v i). On the event {|Tn,i | ≤ ξ, |Tn,i | ≤ ξ}, we use the elementary inequality | log(1 + x)| ≤ 2|x| for all |x| ≤ ξ to derive u v u v | log(1 + Tn,i ) log(1 + Tn,i )| ≤ 4 |Tn,i ||Tn,i |.

It follows from the Cauchy-Schwarz inequality and Theorem 3.1 that   u v u v E log(1 + Tn,i ) log(1 + Tn,i ); |Tn,i | ≤ ξ, |Tn,i | ≤ ξ = O(n−1 ). ii). By Lemma 6.1, we have   u v u v E log(1 + Tn,i ) log(1 + Tn,i ) ; |Tn,i | > ξ, |Tn,i |≤ξ u u ≤ log 2 E[| log(1 + Tn,i )|; |Tn,i | > ξ]

= o(n−1 ). iii). As in ii), we have u v u v E[log(1 + Tn,i ) log(1 + Tn,i ); |Tn,i | ≤ ξ, |Tn,i | > ξ] v v ≤ log 2 E[| log(1 + Tn,i )|; |Tn,i | > ξ]

= o(n−1 ).

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

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iv). By Lemma 6.1 and the Cauchy-Schwarz inequality, we have   u v u v ) log(1 + Tn,i ) ; |Tn,i | > ξ, |Tn,i |>ξ E log(1 + Tn,i   1/2   1/2 u u v v ≤ E log2 (1 + Tn,i ); |Tn,i | > ξ] E log2 (1 + Tn,i ); |Tn,i |>ξ = o(n−1 ). By combining i)-(iv) above, we see that I = O(n−1 ).

(6.11)

By (6.10) and (6.11), we have   (6.12) E (ˆ αii − αii )2 = O(n−1 ) + O(n−2βii ).   Next, we study the cross term E (ˆ α11 −α11 )(ˆ α22 −α22 ) which can be written as   E (α ˆ 11 − α11 )(ˆ α22 − α22 ) m m X X   u v Lu,1 Lv,2 E log(1 + Tn,1 ) log(1 + Tn,2 ) = u=1 v=1 m m X X

+ + +

u=1 v=1 m X m X u=1 v=1 m X m X

  u v Lu,1 Lv,2 E log(1 + Tn,1 ) (log EZ¯n,2 − α22 log v)
  u v Lu,1 Lv,2 log EZ¯n,1 − α11 log u E log(1 + Tn,2 ) u Lu,1 Lv,2 log EZ¯n,1 − α11 log u




v log EZ¯n,2 − α22 log v




u=1 v=1

, I + II + III + IV.

(6.13)   Applying the similar arguments as in evaluating E (ˆ αii − αii )2 , we obtain II = O(n−1−β22 ),

III = O(n−1−β11 ),

IV = O(n−β11 −β22 ).

(6.14)

In order to bound the term I, we distinguish two cases. 1). If (α11 + α22 )/2 = α12 , then by a similar argument as that for proving (6.11) (using Lemma 6.1 and Theorem 3.1) and the fact that Φ0,12 6= 0, we have m X m X

  u v Lu,1 Lv,2 E log(1 + Tn,1 ) log(1 + Tn,2 ) = O(n−1 ).

u=1 v=1

This, together with (6.13) and (6.14), implies   E (ˆ α11 − α11 )(ˆ α22 − α22 ) = O(n−1 ) + O(n−1−β11 ) + O(n−1−β22 ) + O(n−β11 −β22 ).

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

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2). If (α11 + α22 )/2 < α12 , then by an argument similar to that for proving (6.11) and the fact that Φ0,12 = 0, we obtain m X m X

  u v Lu,1 Lv,2 E log(1 + Tn,1 ) log(1 + Tn,2 ) = o(n−1 ).

u=1 v=1

Consequently,   E (ˆ α11 − α11 )(ˆ α22 − α22 ) = o(n−1 ) + O(n−1−β11 ) + O(n−1−β22 ) + O(n−β11 −β22 ). Therefore, we have proved (3.15) and (3.16). u Proof of Theorem 3.5. Recall (3.17) for EZ¯n,i and denote

τ = (τ1,1 , ..., τm,1 , τ1,2 , ..., τm,2 )> . √ ¯ n − τ ) → 0 as n → ∞. By Theorem Since βii > 1/2 for i = 1, 2, we have n(EZ 3.2 and Slutsky’s theorem, we conclude that d ¯n − τ ) − n1/2 (Z → N2m (0, Φ0 ), as n → ∞.

(6.15)

Define a mapping f : R2m → R2 , ∀x , (x1,1 , ..., xm,1 , x1,2 , ..., xm,2 ) ∈ R2m , f (x) :=

m X

Lu,1 log xu,1 ,

u=1

m X

Lu,2 log xu,2

>

.

u=1

¯ n ) and ˆ = f (Z Hence, it is easy to check that f (·) is continuously differentiable, α α = f (τ ). By applying the multivariate delta method (Lehmann and Casella, 2006, Theorem 8.22), we conclude that √

d

ˆ − α) − n(α → N (0, Of (τ )> Φ0 Of (τ )),

e >, L e > )> . where Of (τ ) = (L 1 2 Proof of Lemma 6.1. By H¨ older’s inequality, we have   u k u E log(1 + Tn,i ) ; |Tn,i |>ξ   1/2 1/2  u u ≤ E log2k (1 + Tn,i ) P(|Tn,i | > ξ) .

(6.16)

u First, we establish an upper bound for P(|Tn,i | > ξ). By (3.3), we see that u u the covariance matrix of the random vector Yn,i = (Yn,i (1), ..., Yn,i (n))> is uu n n Σn,i = (σn,ii (j − k))j,k=1 . Let Λn,i = diag(λj,i )j=1 be the diagonal matrix whose diagonal entries are the eigenvalues of Σn,i , and let U = (U1 , ..., Un ) iid

where Uj ∼ N (0, 1), j = 1, ..., n. Then we have 1 > d 1 u Z¯n,i = Yn,i Yn,i = U> Λn,i U. n n

(6.17)

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

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u Since nEZ¯n,i = EU> Λn,i U = tr(Λn,i ), we apply the Hanson and Wright’s inequality (Hanson and Wright, 1971) on tail probability of quadratic forms to obtain

u P |Tn,i | > ξ = P |U> Λn,i U − tr(Λn,i )| > tr(Λn,i ξ)    (tr(Λn,i ))2 tr(Λn,i ) , C6 ξ 2 , (6.18) ≤ exp − min C5 ξ kΛn,i k2 kΛn,i k2F

where kΛn,i k2 and kΛn,i kF are the l2 norm and Frobenius norm of Λn,i respectively and C5 , C6 are positive constants independent of Λn,i , n and ξ. Notice that n X

kΛn,i k2F = tr(Λ2n,i ) = tr(Σ2n ) =

uu (σn,ii (j − k))2 ,

j=1,k=1

and 2 ¯u φuu n,ii = Var(Zn,ii ) = 2 n

n X

uu (σn,ii (j − k))2 .

j=1,k=1

By Theorem 3.1, we have kΛn,i k2F =

n2 uu φ  n φuu 0,ii . 2 n,ii

(6.19)

By combining the above with the facts that kΛn,i k2 ≤ kΛn,i kF and tr(Λn,i )/n = EZ¯ u → Ci uαii as n → ∞, we have n,i

√ tr(Λn,i ) √ tr(Λn,i )/n −1/2 √ & uαii (φuu = n n, 0,ii ) kΛn,i k2 kΛn,i k2 / n and (tr(Λn,i ))2 −1  u2αii (φuu n, as n → ∞. 0,ii ) kΛn,i k2F Hence, when n → ∞, (6.18) decays exponentially with rate when n is large enough, αii

uu

−1/2

√

n. Consequently,

√

u ξ n P(|Tn,i | > ξ) ≤ e−C0 u (φ0,ii ) . (6.20)   u Next, we prove E log2k (1 + Tn,i ) is bounded by C0 n. It is easy to see that       u u u E log2k (1 + Tn,i ) ≤ 22k−1 E log2k Z¯n,i + log2k EZ¯n,i . (6.21)

For any fixed k ∈ Z+ , there exists ck > 1 such that log2k x ≤ x2 , ∀x > ck . Using u the fact that EZ¯n,i → Ci uαii and nΦn → Φ0 as n → ∞, we obtain that for all n large enough, u u u 2 u 2 u E(log2k Z¯n,i ; Z¯n,i > ck ) ≤ E(Z¯n,i ) = (EZ¯n,i ) + Var(Z¯n,i )  u2αii .

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

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u u So the problem is reduced to prove E(log2k Z¯n,i ; Z¯n,i ≤ ck ) ≤ C0 n. It is sufficient to show
u u E log2k Z¯n,i ; Z¯n,i ≤ 1 ≤ C0 n. 2 Let Umin = min1≤i≤n Ui2 . Then n

X tr(Λn,i ) 2 1 u d 1 2 2 λj,i Uj2 ≥ Umin ≥ Ci uαii Umin , C Umin . Z¯n,i = n j=1 n 2

(6.22)

where the second inequality holds for n large enough. 2 Let fn (x) be the density function of Umin , that is ∀x > 0, Z ∞ n−1  2 1 n e−x/2 2 √ √ e−y /2 dy . fn (x) = √ 2πx 2π x It is easy to check that fn (x) ≤

√n . 2πx


u u E log2k Z¯n,i ; Z¯n,i ≤1 ≤E Z C1 log2k (Cx)fn (x)dx ≤ = 0

It follows from (6.22) that
2
2 log2k CUmin ; Umin ≤ 1/C √ Z n C 1 −1 √ y 2 log2k ydy = C0 n, 2π 0

for all n large enough. Therefore we have proven   u E log2k (1 + Tn,i ) ≤ C0 n.

(6.23)

By (6.16), (6.20) and (6.23), we obtain that when n is large, √ √   u k u E log(1 + Tn,i ) ; |Tn,i | > ξ ≤ C7 n1/2 e−C8 ξ n ≤ C7 e−C9 ξ n ,

(6.24)

where C7 , C8 , C9 are independent of n and ξ and C9 < C8 . 7. Appendix A. Remark on Condition (A1). Let F11 , F22 and F12 be the corresponding spectral measure of C11 (·), C22 (·) and C12 (·). By (1.2) and the Tauberian Theorem (see, e.g., Stein (1999)), we have that as x → ∞, Fij (x, ∞) ∼ Cij (0) − Cij (1/x) ∼ e cij |x|−αij , i, j = 1, 2, where e cii = cii for i = 1, 2, and e c12 = c12 ρσ1 σ2 . According to Cramer’s theorem (Yaglom (1987) p.315, Chiles and Delfiner (2009) and Wackernagel (2003)), a necessary and sufficient condition for the matrix (1.1) to be a valid covariance function for X(t) is (F12 (B))2 ≤ F11 (B)F22 (B),

∀B ∈ B(R).

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

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Hence, it is necessary to assume the following conditions on the parameters αij , cij , σi (i = 1, 2) and ρ: α11 + α22 < α12 , or 2 α11 + α22 = α12 and c212 ρ2 σ12 σ22 ≤ c11 c22 . 2

(7.1)

This shows that (1.3) is only slightly stronger than (7.1) in the second case which guarantees that the bivariate process X is not degenerate and satisfies (7.2) below. B. Proof of (1.4). In order to apply Theorem 2.1 in Xiao (1995) to prove (1.4), it is sufficient to verify that there is a constant c > 0 such that
detCov X(s) − X(t) ≥ c |s − t|α11 +α22 (7.2) for all s, t ∈ [0, 1] with |s − t| small enough. Here detCov(ξ) denotes the determinant of the covariance matrix of the random vector ξ. Under Condition (A1), we see that for i = 1, 2,   E (Xi (s) − Xi (t))2 = 2Cii (0) − 2Cii (s − t) ∼ 2cii |s − t|αii ,   E (X1 (s) − X1 (t))(X2 (s) − X2 (t)) = 2C12 (0) − 2C12 (s − t) ∼ 2c12 ρσ1 σ2 |s − t|α12 as |s − t| → 0. Consequently,
detCov X(s) − X(t) ∼ 4 c11 c22 |s − t|α11 +α22 − 4 c212 ρ2 σ12 σ22 |s − t|2α12 . This implies (7.2) and hence proves (1.4). C. Checking the condition (A2) for the bivariate Mat´ern process. Without loss of generality, assume that a = 1 and Mν (h) := M (h|ν, 1). Denote by κν = 21−ν −1 κν . Recall that the derivative of the Bessel Γ(ν) . It satisfies κν+1 = (2ν) function of the second kind Kν satisfies the following recurrence formula (see, e.g., Abramowitz and Stegun (1972), Section 9.6). Kν0 (z) = −Kν+1 (z) +

ν Kν (z), z

(7.3)

and for l ∈ Z, when l < ν < l + 1, we have the following expansion for M (·) M (h|ν, a) =

l X

bj h2j − b|h|2ν + o(|t|2l+2 ),

j=0

where b0 , ..., bl are constants and b = Hence,

Γ(1−ν) 22ν Γ(1+ν)

(see, e.g., Stein (1999), p. 32).

Mν0 (h) = sgn(h)(κν ν|h|ν−1 Kν (|h|) + κν |h|ν Kν0 (|h|))

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

20

= 2ν · sgn(h)|h|−1 (Mν (h) − Mν+1 (h)) = −2νb · sgn(h)|h|2ν−1 + o(|h|2ν−1 ),

(7.4)

where sgn(h) is the sign function. Similarly, 0 Mν00 (h) = (2ν − 1)sgn(h)|h|−1 Mν0 (h) − 2ν · sgn(h)|h|−1 Mν+1 (h)

= −2ν(2ν − 1)b · sgn2 (h)|h|2ν−2 + o(|h|2ν−2 ), 00 Mν(3) (h) = (2ν − 2)sgn(h)|h|−1 Mν00 (h) − 2ν · sgn(h)|h|−1 Mν+1 (h),

= −2ν(2ν − 1)(2ν − 2)b · sgn3 (h)|h|2ν−3 + o(|h|2ν−3 ), ··· Mν(q) (h)

(q−1)

= (2ν − q + 1)sgn(h)|h|−1 Mν(q−1) (h) − 2ν · sgn(h)|h|−1 Mν+1 (h) =−

b(2ν)! sgnq (h)|h|2ν−q + o(|h|2ν−q ). (2ν − q)!

(7.5)

When q = 4, the non-smooth bivariate Mat´ern field {X(t), t ∈ R} satisfies the regularity condition (A2). D. Proof of Theorems 3.1 ∼ 3.2. For proving Theorem 3.1 and Theorem 3.2, we’ll make use of the following lemma. Lemma 7.1. If Conditions (A1) and (A2) hold, then as |h| → ∞, uv σn,ii (h) = O(|h|αii −4 ), uniformly for n > |h|, i = 1, 2

(7.6)

and uv σn,12 (h) = n

α11 +α22 2

−α12

O(|h|α12 −4 ) = O(|h|

α11 +α22 2

−4

)

(7.7)

uniformly for n > |h|. We postpone the proof of Lemma 7.1 to the end of this section. Proof of Theorem 3.1. Let    1− duv n,ij (h) :=  0,

|h| n



uv (σn,ij (h))2 , |h| < n

(7.8)

otherwise.

uv By (3.3) and (3.5), for any fixed h, we have duv n,ij (h) → σ0,ij (h) as n → ∞. By Lemma 7.1, we know αii +αjj −8 duv , n,ij (h) ≤ C0 |h| P∞ with the power αii +αjj −8 < −4. So h=−∞ duv n,ij (h) is bounded by a summable series and (3.11) can be concluded by dominated convergence theorem.

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

21

Proof of Theorem 3.2. The argument in the following generalizes Kent and Wood (1995)’s method to the bivariate case. By the Cram´er-Wold theorem, it is equivalent to prove that for ∀γ = (γ1,1 , ..., γm,1 , γ1,2 , ..., γm,2 )> ∈ R2m , d ¯ n − E[Z ¯ n ]) − n1/2 γ > (Z → N (0, γ > Φ0 γ), as n → ∞.

(7.9)

Let γ i := (γ1,i , ..., γm,i )> , i = 1, 2 and > > > > Γn = diag(γ > , γ > , ..., γ > 1 , γ 2 , γ 2 , ..., γ 2 ) . } | {z } | 1 1{z n times

n times

So Γn is a (2mn)×(2mn) matrix including n copies of γ 1 and γ 2 on the diagonal. Let 1 2 m Yn,i (j) := (Yn,i (j), Yn,i (j), ..., Yn,i (j))> , i = 1, 2, j = 1, 2, ..., n,

and > > > > > > Wn = (Yn,1 (1), Yn,1 (2), ..., Yn,1 (n), Yn,2 (1), Yn,2 (2), ..., Yn,2 (n))> .

So Wn is a (2mn)-dimensional vector. Then, we have ¯ n − E[Z ¯ n ]) = n−1/2 (Wn> Γn Wn − E[Wn> Γn Wn ]). Sn , n1/2 γ > (Z

(7.10)

1

Denote by Vn = E[Wn Wn> ] the covariance matrix of Wn and Vn2 the Cholesky 1

1

factor of Vn , i.e., the lower triangular matrix satisfying Vn = Vn2 (Vn2 )> . Denote by Λn = diag(λn,j )2mn j=1 the diagonal matrix whose diagonal entries 1

1

1

are eigenvalues of 2n− 2 (Vn2 )> Γn Vn2 . Then for a (2mn)-dimensional vector n = (1,n , ..., 2mn,n )> of i.i.d. standard normal random variables, we obtain 1
1 1 d 1 d − 12 n− 2 Wn> Γn Wn = > (Vn2 )> Γn Vn2 n = > Λ n n . n n 2 n

So for ∀θ < min1≤j≤2mn λ−1 n,j , the cumulant generating function Sn is given by kn (θ) , log E[e

θSn

2mn 1X ]=− (log(1 − θλn,j ) + θλn,j ) 2 j=1

(7.11)

To obtain the limit of kn (θ) as n → ∞, we first prove tr(Λ4n ) =

2mn X

λ4n,j → 0, as n → ∞.

j=1

For 1 ≤ i1 , i2 ≤ 2, 1 ≤ j1 , j2 ≤ n, 1 ≤ k1 , k2 ≤ m, let l1 = (i1 − 1)mn + (j1 − 1)m + k1 , l2 = (i2 − 1)mn + (j2 − 1)m + k2 .

(7.12)

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

22

The (l1 , l2 ) entry of Wn is k1 k2 k2 k1 (j2 − j1 ). (j2 )] = σn,i (j1 )Yn,i Vn (l1 , l2 ) = E[Yn,i 1 i2 2 1

(7.13)

So tr(Λ4n ) = =

=

16 n2 16 n2

16 tr((Vn Γn )4 ) n2 2mn X (Vn Γn )(l1 , l2 )(Vn Γn )(l2 , l3 )(Vn Γn )(l3 , l4 )(Vn Γn )(l4 , l1 )

l1 ,l2 ,...,l4 =1 2 X

m X

γk1 ,i1 γk2 ,i2 γk3 ,i3 γk4 ,i4 ∆n (k1 , ..., k4 , i1 , ..., i4 ),

(7.14)

i1 ,...,i4 =1 k1 ,...,k4 =1

where ∆n (k1 , ..., k4 , i1 , ..., i4 ) n X k1 k2 k2 k3 k3 k4 k4 k1 := σn,i (j2 − j1 )σn,i (j3 − j2 )σn,i (j4 − j3 )σn,i (j4 − j1 ). 4 i1 1 i2 2 i3 3 i4 j1 ,...,j4 =1

Letting hi = ji+1 − ji , i = 1, 2, 3, we have ∆n (k1 , ..., k4 , i1 , ..., i4 ) n X k1 k2 k2 k3 k3 k4 k4 k1 = σn,i (h1 )σn,i (h2 )σn,i (h3 )σn,i (h1 + h2 + h3 ). 4 i1 1 i2 2 i3 3 i4 j1 ,...,j4 =1

Given h1 , h2 and h3 fixed, the cardinality of the set #{(j1 , j2 , ..., j4 ) | 1 ≤ j1 , ..., j4 ≤ n} ≤ n. Hence, |∆n (k1 , ..., k4 , i1 , ..., i4 )| X k3 k4 k4 k1 k2 k3 k1 k2 (h1 + h2 + h3 )| (h3 )σn,i ≤n |σn,i (h1 )σn,i (h2 )σn,i 2 i3 3 i4 4 i1 1 i2 |h1 |,|h2 |,|h3 |≤n−1

Further, by Lemma 7.1, we have |∆n (k1 , ..., k4 , i1 , ..., i4 )| ≤C0 n

3 Y

n−1 X

αi i r r 2

hr

+

αi r+1 ir+1 2

−4

r=1 hr =−n+1

≤C0 n

3 ∞ Y X

αi i r r 2

hr

+

αi r+1 ir+1 2

−4

r=1 hr =−∞

=O(n).

(7.15)

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

The last equality holds since we have

αir ir 2

αir+1 ir+1 2

+

23

− 4 < −2. By (7.14) and (7.15),

tr(Λ4n ) = O(n−1 ) → 0, as n → ∞. Now we are ready to prove the asymptotic normality of Sn . Applying Taylor’s expansion to log(1 − θλn,j ) at θ = 0, we obtain kn (θ) =

2mn 2mn 2mn θ3 X 3 θ4 X θ2 X 2 λn,j + λn,j + (1 − θn,j λn,j )−4 λ4n,j , 4 j=1 6 j=1 8 j=1

where θn,j is between 0 and θ. Let us first consider the term

1 2

P2mn j=1

(7.16)

λ2n,j . Since

2mn 1X 2 1 2 λ = tr(Λ2n ) = tr((Vn Γn )2 ) 2 j=1 n,j 2 n

=

2 m 2 X X n i ,i =1

n X

2 X

m X

1

2


2 k1 k2 (j2 − j1 ) , γk1 ,i1 γk2 ,i2 σn,i 1 i2

k1 ,k2 =1 j1 ,j2 =1

and γ > Φn γ =

1 k2 γk1 ,i1 γk2 ,i2 φkn,i 1 i2

i1 ,i2 =1 k1 ,k2 =1

=

2 n2

2 X

m X

n X


2 k1 k2 γk1 ,i1 γk2 ,i2 σn,i (j2 − j1 ) , 1 i2

i1 ,i2 =1 k1 ,k2 =1 j1 ,j2 =1

it follows from Theorem 3.1 that 2mn 1X 2 λ = γ > (nΦn )γ → γ > Φ0 γ, as n → ∞. 2 j=1 n,j

(7.17)

Secondly, by (7.12), we have max

1≤j≤2mn

|λn,j | ≤

 2mn X

λ4n,j

 14 → 0, as n → ∞,

(7.18)

j=1

which implies 2mn 2mn X X 3 ≤ max |λn,j | λ λ2n,j → 0, as n → ∞. n,j 1≤j≤2mn j=1

(7.19)

j=1

Thirdly, notice that δ := supn≥1 max1≤j≤2mn |λn,j | is positive and finite by (7.18). If we restrict attention to |θ| ≤ 12 δ −1 , we have (1 − θn,j λn,j )−4 ≤ 16 and

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

24

hence for θ ∈ (− 12 δ −1 , 12 δ −1 ), 2mn X

(1 − θn,j λn,j )−4 λ4n,j → 0, as n → ∞.

(7.20)

j=1

Therefore, by (7.17),(7.19) and (7.20), for ∀θ ∈ (− 12 δ −1 , 12 δ −1 ), we have θ2 > γ Φ0 γ, 2

kn (θ) → which leads to

d ¯ n − E[Z ¯ n ]) − Sn := n1/2 γ > (Z → N (0, γ > Φ0 γ), as n → ∞.

This proves Theorem 3.2. Finally, we prove Lemma 7.1. Proof of Lemma 7.1. (7.6) comes directly from the proof of Theorem 1 in Kent
and Wood (1997). We only need to prove (7.7). To this end, we expand C12 h+kv−ju n in a Taylor series about h/n to the fourth order to obtain uv σn,12 (h) = n

1 X

α11 +α22 2

 aj ak C12

j,k=−1

=n

α11 +α22 2

1 3 X X

aj ak

r=0 j,k=−1

+n

α11 +α22 2

1 X

aj ak

j,k=−1

=n

α11 +α22 2

1 X j,k=−1

aj ak

h + kv − ju n



  (kv − ju)r (r) h C 12 r!nr n

  (kv − ju)4 (4) h∗kj C 12 4!n4 n

  (kv − ju)4 (4) h∗kj C , 12 4!n4 n

(7.21)

where h∗kj lies between h and h + kv − ju. Since |kv − ju| ≤ u + v ≤ 2m, h∗kj ≤ 2|h| for all |h| ≥ 2m. By applying condition (A2) to the last terms in (7.21), we derive that uv |σn,12 (h)| ≤ C0 |h|α12 −4 · n

α11 +α22 2

−α12

≤ C0 |h|

for all |h| ≥ 2m and all n > |h|. This concludes the proof.

α11 +α22 2

−4

Y. Zhou and Y. Xiao/Joint Asymptotics for Bivariate Fractal Indices

25

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