Scaling transition and edge effects for negatively dependent linear random fields on Z2

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Abstract

We obtain a complete description of anisotropic scaling limits and the existence of scaling transition for a class of negatively dependent linear random fields X on Z2 with moving-average coefficients a(t,s) decaying as |t|q1 and |s|q2 in the horizontal and vertical directions, q11+q21<1 and satisfying (t,s)Z2a(t,s)=0. The scaling limits are taken over rectangles whose sides increase as λ and λγ when λ, for any γ>0. The scaling transition occurs at γ0X>0 if the scaling limits of X are different and do not depend on γ for γ>γ0X and γ<γ0X. We prove that the scaling transition in this model is closely related to the presence or absence of the edge effects. The paper extends the results in Pilipauskaitė and Surgailis (2017) on the scaling transition for a related class of random fields with long-range dependence.

Introduction

A stationary random field (RF) X={X(t);tZν} on ν-dimensional lattice Zν,ν1 with finite variance is said to be (covariance) long-range dependent (LRD) if tZν|Cov(X(t),X(0))|=, short-range dependent (SRD) if tZν|Cov(X(t),X(0))|<,tZνCov(X(t),X(0))0, and negatively dependent (ND) if tZν|Cov(X(t),X(0))|<,tZνCov(X(t),X(0))=0. The above definitions apply per se to RFs with finite 2nd moment; related albeit not equivalent definitions of LRD, SRD, and ND properties are discussed in [11], [16], [32] and other works. For linear (moving-average) RFs, Lahiri and Robinson [16] define similar concepts through summability properties of the moving-average coefficients. The last paper also discusses the importance of spatial LRD in applied sciences, including the relevant literature.

The above classification plays an important role in limit theorems. Consider the sum SKλXtKλX(t) of the values of RF X over large ‘sampling region’ KλZν with |Kλ|=tKλ1(λ). Under additional conditions, the variance Var(SKλX) grows faster than |Kλ| under LRD, as O(|Kλ|) under SRD, and slower than |Kλ| under ND. In the latter case, Var(SKλX) may grow as slow as the ‘volume’ |Kλ| of the boundary Kλ, or even slower than |Kλ|, giving rise to ‘edge effects’ which may affect or dominate the limit distribution of SKλX; see [16], [33].

Probably, the most studied case of limit theorems for RFs deals with rectangular summation regions, which allows for partial sums and limit RFs, similarly as in the case ν=1. Let X={X(t);tZν} be a stationary random field (RF) on Zν,ν1,γ=(γ1,,γν)R+ν be a collection of positive numbers, and Kλ,γ(x)[1,λγ1x1]××[1,λγνxν],x=(x1,,xν)R+ν,be a family of ν-dimensional ‘rectangles’ indexed by λ>0, whose sides grow at generally different rate O(λγi),i=1,,ν as λ, and Sλ,γX(x)tKλ,γ(x)X(t),xR+νbe the corresponding partial sums RF. [29], [31], [32], [36] discussed the anisotropic scaling limits for any γR+ν of some classes of LRD RFs X in dimension ν=2,3, viz., Aλ,γ1Sλ,γX(x)fddVγX(x),xR+νas λ, where Aλ,γ is a normalization. Following [28], [36] the family {VγX;γR+ν} of all scaling limits in (1.3) will be called the scaling diagram of RF X. [28] noted that the scaling diagram provides a more complete ‘large-scale summary’ of RF X compared to (usual) isotropic or anisotropic scaling at fixed γR+ν as discussed in [2], [7], [16], [17], [19], [35] and other works.

[29], [31], [32] observed that for a large class of LRD RFs X in dimension ν=2, the scaling diagram essentially consists of three elements: VX={V0X,V+X,VX}, V0X termed the well-balanced and V±X the unbalanced scaling limits of X. For ν=2 without loss of generality (w.l.g.) we can assume γ=(1,γ)R+2 or γ1=1,γ2=γ in (1.2) and denote Sλ,γX(x,y)(t,s)K[λx,λγy]X(t,s),(x,y)R+2and VγX,Aλ,γ the corresponding sum in (1.2) and quantities in (1.3) defined for γ>0, where K[λx,λγy]={(t,s)Z2:1tλx,1sλγy}. [29], [31], [32] proved that there exists a (nonrandom) γ0X>0 such that VγX do not depend on γ for γ>γ0X and γ<γ0X, viz., VγX=V+X,γ>γ0X,VX,γ<γ0X,V0X,γ=γ0Xand V+XfddaVX(a>0). The above fact was termed the scaling transition [31], [32]. It was noted in the above-mentioned works that the scaling transition constitutes a new and general feature of spatial dependence which occurs in many spatio-temporal models including telecommunications and economics [9], [14], [18], [23], [25], [26], [27], [28]. However, as noted in [29], [36], these studies were limited to LRD models and the existence of the scaling transition under ND remained open.

The present paper discusses the scaling transition for linear ND RFs on Z2 having a moving-average representation X(t,s)=(u,v)Z2a(tu,sv)ε(u,v),(t,s)Z2,in standardized i.i.d. sequence {ε(u,v);(u,v)Z2},Eε(u,v)=0,Eε2(u,v)=1 with deterministic moving-average coefficients a(t,s)=1(|t|2+|s|2q2q1)q12(L0(t(|t|2+|s|2q2q1)12)+o(1)),|t|+|s|, (t,s)(0,0), where qi>0,i=1,2 satisfy 0<Q1q1+1q2<2which imply qi>12,i=1,2. In (1.7), L0(u),u[1,1] is a bounded piece-wise continuous function on [1,1] termed the angular function in [29]. (We note that the boundedness and continuity assumptions on the angular function do not seem necessary for our results and possibly can be relaxed.) The form of moving-average coefficients in (1.7) is the same as in [29] and can be generalized to some extent but we prefer to use (1.7) for better comparison with the results of [29]. Condition Q<1 guarantees that (t,s)Z2|a(t,s)|< and the ND property of X in (1.6) is a consequence of the zero-sum condition: (t,s)Z2a(t,s)=0,forQ<1.In contrast, [29] assumes 1<Q<2 implying (t,s)Z2|a(t,s)|= and the LRD property of the corresponding linear RF X in (1.6). The linear model in (1.6) and the results of this paper can be regarded as an extension of the classical results of Davydov [6] in the time series setting, who identified all partial sums limits (fractional Brownian motions) of moving-average processes with one-dimensional ‘time’.

The main results of this paper and [29] are illustrated in Fig. 1 showing 8 regions R11,,R33 of the parameter set {(1q1,1q2):0<Q<2} of the linear RF X in (1.6)–(1.9) with different unbalanced limits. We remark that when (t,s)Z2a(t,s)0 and Q<1 the linear RF X in (1.6)–(1.7) is SRD and all scaling limits VγX,γ>0 agree with Brownian sheet B12,12, see ([29], Theorem 3.4), meaning that in this case the scaling transition does not occur.

The regions R11,,R33 in Fig. 1 are described in Table 1. Recall the definition of fractional Brownian sheet (FBS) BH1,H2={BH1,H2(x,y);(x,y)R+2} with Hurst parameters 0<H1,H21 as a Gaussian process with zero mean and covariance EBH1,H2(x1,y1)BH1,H2(x2,y2)=(14)(x12H1+x22H1|x1x2|2H1)(y12H2+y22H2|y1y2|2H2), see [3]. FBS BH1,H2 with one of the parameters Hi equal to 12 or 1 have a very specific dependence structure (either independent or completely dependent (invariant) increments in one direction, see [31], [32]) and play a particular role in our work. We extend the above class of Gaussian RFs to all parameter values (H1,H2)[0,1]2 by setting EBH1,0(x1,y1)BH1,0(x2,y2):=limH2+0EBH1,H2(x1,y1)BH1,H2(x2,y2),0<H11,EB0,H2(x1,y1)B0,H2(x2,y2):=limH1+0EBH1,H2(x1,y1)BH1,H2(x2,y2),0<H21,EB0,0(x1,y1)B0,0(x2,y2):=limH1,H2+0EBH1,H2(x1,y1)BH1,H2(x2,y2) for any (xi,yi)R+2,i=1,2. The limit covariances on the r.h.s. of (1.11) are explicitly written in Remark 2.1. FBS BH1,H2 where one or both parameters are equal to 0 are rather unusual (have nonseparable paths). The appearance of such RFs in limit theorems is surprising but seems to be rather common under edge effects; see Remark 3.2.

Parameters H̃i,Hi,i=1,2 in Table 1 (expressed in terms of q1,q2) are specified in the beginning of Section 2. The description in Table 1 is not very precise since it omits various asymptotic constants which may vanish in some cases, meaning that some additional conditions on a(t,s) are needed for the validity of the results in this table. The rigorous formulations including the normalizing constants Aλ,γ are presented in Section 2. The limit distributions in regions R11,R12,R21,R22+ refer to LRD set-up and are part of [29]. The new results under ND refer to regions R22,R23,R32,R33 of Table 1. Note 2q1(1Q)>q1q2 in R23 and 1(2q2(1Q))<q1q2 in R32. The limit B12,0 in region R23 can be related to the ‘horizontal edge effect’ which dominates the limit distribution of Sλ,γX of (1.4) unless the vertical length O(λγ) of K[λx,λγy] grows fast enough vs. its horizontal length O(λ), or γ>2q1(1Q) holds, in which case FBS BH1,12 dominates. Similarly, B0,12 in region R32 can be related to the ‘vertical edge effect’ appearing in the limit of Sλ,γX unless the vertical length O(λγ) increases sufficiently slow w.r.t the horizontal length O(λ), or γ<12q2(1Q) holds, in which case FBS B12,H2 dominates. Finally, R33 can be characterized as the parameter region where the edge effects (either horizontal, or vertical) completely dominate the limit behavior of Sλ,γX. The above interpretation of R32,R23 and R33 is based on the approximations of Sλ,γX by suitable ‘edge terms’ which are discussed in Section 3.

The results of the present work are related to the work Lahiri and Robinson [16] which discussed the limit distribution of sums of linear LRD, SRD and ND RFs over homothetically inflated or isotropically rescaled (i.e., γ1==γν=1) star-like regions Kλ of very general form. This generality of Kλ does not seem to allow for a natural introducing of partial sums, restricting the problem to the convergence of one-dimensional distributions in contrast to finite-dimensional distributions in the present paper. While [16] considers several forms of moving-average coefficients, the only case when a(t,s) in (1.7) satisfy the assumptions in [16] seems to be the ‘isotropic’ case q1=q2. As explained in Remark 2.3, in the case q1=q2 and γ=(1,1) our limit results agree with [16], including the ‘edge effect’. We also note Damarackas and Paulauskas [5] who discussed partial sums limits of linear LRD, SRD and ND RFs, possibly with infinite variance and moving-average coefficients which factorize along coordinate axes (i.e., different from (1.7)) in which case the scaling limits in (1.3) do not depend on γ and the scaling transition does not exist. See also ([36], Remark 4.1).

The rest of the paper is organized as follows. Section 2 contains the main results (Proposition 2.1, Theorems 2.2, 2.3). The proofs of these results are given in Section 3. Section 4 presents two examples of fractionally integrated ND RFs, extending the examples of fractionally integrated LRD RFs in [29]. Section 5 (Concluding remarks) summarizes the main discoveries and mentions several open problems.

Notation. In what follows, C denote generic positive constants which may be different at different locations. We write fdd,=fdd, and fdd for the weak convergence, equality and inequality of finite-dimensional distributions, respectively. R+ν{x=(x1,,xν)Rν:xi>0,i=1,,ν},R+R+1, xmax{kZ:kx},xmin{kZ:kx},xR. 1(A) stands for the indicator function of a set A.

Section snippets

Main results

Throughout the paper we use the following notation: H132+q1q2q1=12+q1(Q1),H232+q2q1q2=12+q2(Q1), H̃132q1+q12q2=1q12(2Q),H̃232q2+q22q1=1q22(2Q),γ0q1q2,γedge,1012q2(1Q),γedge,202q1(1Q),Qedge,132q1+1q2,Qedge,21q1+32q2,Q˜112q1+1q2,Q˜21q1+12q2. We recall that qi,i=1,2 are the asymptotic parameters of the MA coefficients in (1.7) satisfying (1.9) and Q is defined in (1.8). Hi,H̃i,i=1,2 in (2.1) are the Hurst parameters of the limit FBS in Table 1 and in the subsequent Theorem

Outline of the proof and preliminaries

Let us explain the main steps of the proof of Theorem 2.2, Theorem 2.3, Theorem 2.4. By definition, Sλ,γX(x,y) can be rewritten as a weighted sum Sλ,γX(x,y)=(u,v)Z2ε(u,v)Gλ,γ(u,v),Gλ,γ(u,v)(t,s)K[λx,λγy]a(tu,sv)in i.i.d.r.v.s ε(u,v),(u,v)Z2. It happens that different regions of ‘noise locations’ (u,v) contribute to different limit distributions in our theorems.

The basic decomposition Z2=K((i,j)=1,0,1,(i,j)(0,0)Ki,jc),of Z2 into 9 sets is shown in Fig. 2. Following (3.2) we decompose

Fractionally integrated negative dependent RFs

1. Isotropic fractionally integrated RF. Introduce the (discrete) Laplacian ΔY(t,s)(14)|u|+|v|=1 (Y(t+u,s+v)Y(t,s)) and a lattice isotropic linear RF X(t,s)=(u,v)Z2a(u,v)ε(tu,sv),(t,s)Z2where {ε(t,s),(t,s)Z2} are standard i.i.d. r.v.s, a(u,v)j=0ψj(d)pj(u,v),ψj(d)Γ(jd)Γ(j+1)Γ(d),|d|<12and pj(u,v) are j-step transition probabilities of a symmetric nearest-neighbor random walk {Wj;j=0,1,} on Z2 with equal 1-step probabilities P(W1=(u,v)|W0=(0,0))=14,|u|+|v|=1. Note j=0|ψj(d)

Concluding remarks

The present paper and [29] provide a nearly complete description, summarized in Table 1, of anisotropic scaling limits on inflated rectangles of linear LRD, SRD and ND RFs on Z2 with coefficients decaying at generally different rate in the horizontal and vertical directions. It is proved that under ND these limits may exhibit horizontal and/or vertical edge effects, meaning that the main contribution to the limit comes from values of the RF near the horizontal and/or vertical edges of the

Acknowledgments

The author is grateful to two anonymous referees for useful comments.

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