Anisotropic scaling limits of long-range dependent random fields

Some remarks on scaling transition in limit theorems for random fields

“…i.e., we have the same filter (11) (with d = 2) considered in the previous section. From ( 8)-( 10) and (12) we have that…”

“…In the above mentioned papers [8], [9], and [7] only r.f.s, defined on Z 2 were considered. It was mentioned that generalization of the results of these papers to the case Z d with d ≥ 3 is difficult task, and the first step in this direction was the paper [11], where the scaling transition of linear random fields on Z 3 was considered. But at first we must discuss what we understand by words scaling transition on Z 3 , or more generally, on Z d , d ≥ 3.…”

“…In this case there was quite natural way to define this dependence, using relation n 2 = f (n 1 ) between n 1 and n 2 , see Definitions 2 and 6. But the straightforward generalization of Definition 2, considering sums (7) in the case d = 3 and assuming n 1 = n q 1 , n 2 = n q 2 , n 3 = n q 3 (such case is considered in [11]), is too narrow, since it presents only one possible way to define the path in Z 3 + . Probably for this reason in [11] there is no strict definition of the scaling transition, and the author in [11] wrote "...we do not attempt to provide a formal definition of scaling transition for RFs in dimensions d ≥ 3 since further studies are needed to fully understand it".…”

“…But the straightforward generalization of Definition 2, considering sums (7) in the case d = 3 and assuming n 1 = n q 1 , n 2 = n q 2 , n 3 = n q 3 (such case is considered in [11]), is too narrow, since it presents only one possible way to define the path in Z 3 + . Probably for this reason in [11] there is no strict definition of the scaling transition, and the author in [11] wrote "...we do not attempt to provide a formal definition of scaling transition for RFs in dimensions d ≥ 3 since further studies are needed to fully understand it". Our examples of this section show that in dimension 3 we have much more possibilities (comparing with the case d = 2) to define paths of (n 1 , n 2 , n 3 ) growing to infinity, moreover, with growing dimension the complexity grows very rapidly.…”

“…indexed by indices in Z d , d > 2. At the final stage of the preparation of the paper we became aware of the publication [11], where scaling transition is investigated for linear random fields on Z 3 . Taking into account the results of this paper and our simple examples, one can say that with d increasing the picture of scaling transition becomes more complicated, even for those simple examples of linear random fields which we consider.…”