Scaling transition and edge effects for negatively dependent linear random fields on
Introduction
A stationary random field (RF) on -dimensional lattice with finite variance is said to be (covariance) long-range dependent (LRD) if , short-range dependent (SRD) if , and negatively dependent (ND) if . 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 of the values of RF over large ‘sampling region’ with . Under additional conditions, the variance grows faster than under LRD, as under SRD, and slower than under ND. In the latter case, may grow as slow as the ‘volume’ of the boundary , or even slower than , giving rise to ‘edge effects’ which may affect or dominate the limit distribution of ; 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 . Let be a stationary random field (RF) on be a collection of positive numbers, and be a family of -dimensional ‘rectangles’ indexed by , whose sides grow at generally different rate as , and be the corresponding partial sums RF. [29], [31], [32], [36] discussed the anisotropic scaling limits for any of some classes of LRD RFs in dimension , viz., as , where is a normalization. Following [28], [36] the family of all scaling limits in (1.3) will be called the scaling diagram of RF . [28] noted that the scaling diagram provides a more complete ‘large-scale summary’ of RF compared to (usual) isotropic or anisotropic scaling at fixed as discussed in [2], [7], [16], [17], [19], [35] and other works.
[29], [31], [32] observed that for a large class of LRD RFs in dimension , the scaling diagram essentially consists of three elements: , termed the well-balanced and the unbalanced scaling limits of . For without loss of generality (w.l.g.) we can assume or in (1.2) and denote and the corresponding sum in (1.2) and quantities in (1.3) defined for , where . [29], [31], [32] proved that there exists a (nonrandom) such that do not depend on for and , viz., and . 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 having a moving-average representation in standardized i.i.d. sequence with deterministic moving-average coefficients , where satisfy which imply . In (1.7), is a bounded piece-wise continuous function on 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 guarantees that and the ND property of in (1.6) is a consequence of the zero-sum condition: In contrast, [29] assumes implying and the LRD property of the corresponding linear RF 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 of the parameter set of the linear RF in (1.6)–(1.9) with different unbalanced limits. We remark that when and the linear RF in (1.6)–(1.7) is SRD and all scaling limits agree with Brownian sheet , see ([29], Theorem 3.4), meaning that in this case the scaling transition does not occur.
The regions in Fig. 1 are described in Table 1. Recall the definition of fractional Brownian sheet (FBS) with Hurst parameters as a Gaussian process with zero mean and covariance see [3]. FBS with one of the parameters equal to or 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 by setting for any . The limit covariances on the r.h.s. of (1.11) are explicitly written in Remark 2.1. FBS 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 in Table 1 (expressed in terms of ) 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 are needed for the validity of the results in this table. The rigorous formulations including the normalizing constants are presented in Section 2. The limit distributions in regions refer to LRD set-up and are part of [29]. The new results under ND refer to regions of Table 1. Note in and in . The limit in region can be related to the ‘horizontal edge effect’ which dominates the limit distribution of of (1.4) unless the vertical length of grows fast enough vs. its horizontal length , or holds, in which case FBS dominates. Similarly, in region can be related to the ‘vertical edge effect’ appearing in the limit of unless the vertical length increases sufficiently slow w.r.t the horizontal length , or holds, in which case FBS dominates. Finally, can be characterized as the parameter region where the edge effects (either horizontal, or vertical) completely dominate the limit behavior of . The above interpretation of and is based on the approximations of 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., ) star-like regions of very general form. This generality of 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 in (1.7) satisfy the assumptions in [16] seems to be the ‘isotropic’ case . As explained in Remark 2.3, in the case and 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 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, denote generic positive constants which may be different at different locations. We write , and for the weak convergence, equality and inequality of finite-dimensional distributions, respectively. , . stands for the indicator function of a set .
Section snippets
Main results
Throughout the paper we use the following notation: We recall that are the asymptotic parameters of the MA coefficients in (1.7) satisfying (1.9) and is defined in (1.8). 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, can be rewritten as a weighted sum in i.i.d.r.v.s . It happens that different regions of ‘noise locations’ contribute to different limit distributions in our theorems.
The basic decomposition of 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 and a lattice isotropic linear RF where are standard i.i.d. r.v.s, and are -step transition probabilities of a symmetric nearest-neighbor random walk on with equal 1-step probabilities . Note
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 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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