Scaling transition and edge effects for negatively dependent linear random fields on ${\mathbb{Z}}^2$

TL;DR

This paper characterizes the anisotropic scaling limits and transition phenomena for negatively dependent linear random fields on Z^2, revealing how edge effects influence the structure of these limits as the domain size grows.

Contribution

It provides a complete description of scaling limits and transitions for negatively dependent linear random fields with specific decay rates, highlighting the role of edge effects.

Findings

Complete description of anisotropic scaling limits.

Identification of conditions for scaling transition.

Edge effects significantly influence the structure of limits.

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 on ${\mathbb{Z}}^2$ with moving-average coefficients $a(t,s)$ decaying as $|t|^{-q_1}$ and $|s|^{-q_2}$ in the horizontal and vertical directions, $q_1^{-1} + q_2^{-1} < 1 $. The scaling limits are taken over rectangles whose sides increase as $\lambda $ and $\lambda^\gamma $ when $\lambda \to \infty$, for any $\gamma >0$. We prove that the scaling transition %and the structure of the scaling diagram in this model is closely related to the presence or absence of the edge effects.

Figures

Tables

Table 1. Table 1: Unbalanced scaling limits V ± 𝔛 subscript superscript 𝑉 𝔛 plus-or-minus V^{\mathfrak{X}}_{\pm} in regions R 11 , ⋯ , R 33 subscript 𝑅 11 ⋯ subscript 𝑅 33 R_{11},\cdots,R_{33} in Fig. 1

Parameter region	Critical ${\gamma }_0^{𝔛}$	$V_{+}^{𝔛}$	$V_{-}^{𝔛}$	Range of Hurst parameters
$R_{11}$	$q_1/q_2$	$B_{1,{\tilde{H}}_2}$	$B_{{\tilde{H}}_1,1}$
$R_{12}$	$q_1/q_2$	$B_{H_1,1/2}$	$B_{{\tilde{H}}_1,1}$
$R_{21}$	$q_1/q_2$	$B_{1,{\tilde{H}}_2}$	$B_{1/2,H_2}$
$R_{22}^{+}$	$q_1/q_2$	$B_{H_1,1/2}$	$B_{1/2,H_2}$
$R_{22}^{-}$	$q_1/q_2$	$B_{H_1,1/2}$	$B_{1/2,H_2}$
$R_{23}$	$2q_1\left(1-Q\right)$	$B_{H_1,1/2}$	$B_{1/2,0}$
$R_{32}$	$1/2q_2\left(1-Q\right)$	$B_{0,1/2}$	$B_{1/2,H_2}$
$R_{33}$	$1$	$B_{0,1/2}$	$B_{1/2,0}$

Equations

$$K_{\lambda ,\mathbf{\gamma }}(\mathbf{x})$$

$$S_{\lambda ,\mathbf{\gamma }}^{\mathfrak{X}}(\mathbf{x})$$

$$A^{-1}_{\lambda,\boldsymbol{\gamma}}S^{\mathfrak{X}}_{\lambda,\boldsymbol{\gamma}}(\boldsymbol{x})\ \begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\longrightarrow}}\\ \end{array}\ V^{\mathfrak{X}}_{\boldsymbol{\gamma}}(\boldsymbol{x}),\quad\boldsymbol{x}\in\mathbb{R}^{\nu}_{+}$$

$$S_{\lambda ,\gamma }^{\mathfrak{X}}(x,y):=\sum _{(t,s)\in K_{[\lambda x,{\lambda }^{\gamma }y]}}\mathfrak{X}(t,s),(x,y)\in {\mathbb{R}}_{+}^2$$

$$V_{\gamma }^{\mathfrak{X}}=\{\begin{array}{ll}{V}_{+}^{\mathfrak{X}},& \gamma >{\gamma }_0^{\mathfrak{X}},\\ V_{-}^{\mathfrak{X}},& \gamma <{\gamma }_0^{\mathfrak{X}},\\ V_0^{\mathfrak{X}},& \gamma ={\gamma }_0^{\mathfrak{X}}\end{array}$$

$$\mathfrak{X}(t,s)=\sum _{(u,v)\in {\mathbb{Z}}^2}a(t-u,s-v)\epsilon (u,v),(t,s)\in {\mathbb{Z}}^2,$$

$$a(t,s)\ =\ \frac{1}{(|t|^{2}+|s|^{2q_{2}/q_{1}})^{q_{1}/2}}\Big{(}L_{0}\big{(}\frac{t}{(|t|^{2}+|s|^{2q_{2}/q_{1}})^{1/2}}\big{)}+o(1)\Big{)},\qquad|t|+|s|\to\infty,$$

$$0<Q:=\frac{1}{q_1}+\frac{1}{q_2}<2$$

$$\sum _{(t,s)\in {\mathbb{Z}}^2}a(t,s)$$

$$\mathrm{E}{B}_{H_1,H_2}(x_1,y_1)B_{H_1,H_2}(x_2,y_2)=(1/4)(x_1^{2H_1}+x_2^{2H_1}-|x_1-x_2{|}^{2H_1})(y_1^{2H_2}+y_2^{2H_2}-|y_1-y_2{|}^{2H_2}),$$

$$\mathrm{E}{B}_{H_1,0}(x_1,y_1)B_{H_1,0}(x_2,y_2)$$

$$\mathrm{E}{B}_{0,H_2}(x_1,y_1)B_{0,H_2}(x_2,y_2)$$

$$\mathrm{E}{B}_{0,0}(x_1,y_1)B_{0,0}(x_2,y_2)$$

$$H_1:=\frac{3}{2}+\frac{q_1}{q_2}-q_1=\frac{1}{2}+q_1(Q-1),H_2:=\frac{3}{2}+\frac{q_2}{q_1}-q_2=\frac{1}{2}+q_2(Q-1),$$

$${\tilde{H}}_1:=\frac{3}{2}-q_1+\frac{q_1}{2q_2}=1-\frac{q_1}{2}(2-Q),{\tilde{H}}_2:=\frac{3}{2}-q_2+\frac{q_2}{2q_1}=1-\frac{q_2}{2}(2-Q),$$

$${\gamma }^0:=\frac{q_1}{q_2},{\gamma }_{\mathrm{edge},1}^0:=\frac{1}{2q_2(1-Q)},{\gamma }_{\mathrm{edge},2}^0:=2q_1(1-Q),$$

$$Q_{\mathrm{edge},1}:=\frac{3}{2q_1}+\frac{1}{q_2},Q_{\mathrm{edge},2}:=\frac{1}{q_1}+\frac{3}{2q_2},{\tilde{Q}}_1:=\frac{1}{2q_1}+\frac{1}{q_2},{\tilde{Q}}_2:=\frac{1}{q_1}+\frac{1}{2q_2}.$$

$$a_{\infty }(t,s)$$

$$h_0(x,y;u,v)$$

$$h_1(x,y;u,v)$$

$$h_2(x,y;u,v)$$

$$h_0(x,y;u,v)$$

$$h_1(x,y;u,v)$$

$$h_2(x,y;u,v)$$

$${\tilde{h}}_1(x,y;u,v):=x{\int }_{(0,y]}{a}_{\infty }(u,s-v)\mathrm{d}s,{\tilde{h}}_2(x,y;u,v):=y{\int }_{(0,x]}{a}_{\infty }(t-u,v)\mathrm{d}t$$

$$V_0^{\mathfrak{X}}(x,y)$$

$$V_{+}^{\mathfrak{X}}(x,y)$$

$$V_{-}^{\mathfrak{X}}(x,y)$$

$$V_{+}^{\mathfrak{X}}(x,y)$$

$$V_{-}^{\mathfrak{X}}(x,y)$$

$$V_{+}^{\mathfrak{X}}$$

$$V_{-}^{\mathfrak{X}}$$

$$h_1(x,y;u,v)$$

$$h_2(x,y;u,v)$$

$$L_{1,\pm }:={\int }_{\mathbb{R}}{a}_{\infty }(\pm 1,s)\mathrm{d}s,L_{2,\pm }:={\int }_{\mathbb{R}}{a}_{\infty }(t,\pm 1)\mathrm{d}t$$

$$h_1(x,y;u,v)$$

Full text

Scaling transition and edge effects for negatively

dependent

linear random fields on ${\mathbb{Z}}^{2}$

Donatas Surgailis 111E-mail: donatas.surgailis@mif.vu.lt

*Vilnius University, Faculty of Mathematics and Informatics, Naugarduko 24, 03225 Vilnius, Lithuania *

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 $\mathfrak{X}$ on ${\mathbb{Z}}^{2}$ with moving-average coefficients $a(t,s)$ decaying as $|t|^{-q_{1}}$ and $|s|^{-q_{2}}$ in the horizontal and vertical directions, $q_{1}^{-1}+q_{2}^{-1}<1$ and satisfying $\sum_{(t,s)\in\mathbb{Z}^{2}}a(t,s)=0$. The scaling limits are taken over rectangles whose sides increase as $\lambda$ and $\lambda^{\gamma}$ when $\lambda\to\infty$, for any $\gamma>0$. The scaling transition occurs at $\gamma^{\mathfrak{X}}_{0}>0$ if the scaling limits of $\mathfrak{X}$ are different and do not depend on $\gamma$ for $\gamma>\gamma^{\mathfrak{X}}_{0}$ and $\gamma<\gamma^{\mathfrak{X}}_{0}$. 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.

1 Introduction

A stationary random field (RF) $\mathfrak{X}=\{\mathfrak{X}({\boldsymbol{t}});{\boldsymbol{t}}\in\mathbb{Z}^{\nu}\}$ on $\nu$-dimensional lattice $\mathbb{Z}^{\nu},\nu\geq 1$ with finite variance is said to be (covariance) long-range dependent (LRD) if \sum_{\boldsymbol{t}\in\mathbb{Z}^{\nu}}|{\rm Cov}(\mathfrak{X}({\boldsymbol{t}}),\mathfrak{X}({\mbox{\boldmath0}}))|=\infty,  short-range dependent (SRD) if  \sum_{{\boldsymbol{t}}\in\mathbb{Z}^{\nu}}|{\rm Cov}(\mathfrak{X}({\boldsymbol{t}}),\mathfrak{X}({\mbox{\boldmath0}}))|<\infty,\ \sum_{{\boldsymbol{t}}\in\mathbb{Z}^{\nu}}{\rm Cov}(\mathfrak{X}({\boldsymbol{t}}),\mathfrak{X}({\mbox{\boldmath0}}))\neq 0, and negatively dependent (ND) if  \sum_{{\boldsymbol{t}}\in\mathbb{Z}^{\nu}}|{\rm Cov}(\mathfrak{X}({\boldsymbol{t}}),\mathfrak{X}({\mbox{\boldmath0}}))|<\infty,\ \sum_{{\boldsymbol{t}}\in\mathbb{Z}^{\nu}}{\rm Cov}(\mathfrak{X}({\boldsymbol{t}}),\mathfrak{X}({\mbox{\boldmath0}}))=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 [12, 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 $S^{\mathfrak{X}}_{K_{\lambda}}:=\sum_{{\boldsymbol{t}}\in K_{\lambda}}\mathfrak{X}({\boldsymbol{t}})$ of the values of RF $\mathfrak{X}$ over large ‘sampling region’ $K_{\lambda}\subset\mathbb{Z}^{\nu}$ with $|K_{\lambda}|=\sum_{{\boldsymbol{t}}\in K_{\lambda}}1\to\infty\ (\lambda\to\infty)$. Under additional conditions, the variance ${\rm Var}(S^{\mathfrak{X}}_{K_{\lambda}})$ grows faster than $|K_{\lambda}|$ under LRD, as $O(|K_{\lambda}|)$ under SRD, and slower than $|K_{\lambda}|$ under ND. In the latter case, ${\rm Var}(S^{\mathfrak{X}}_{K_{\lambda}})$ may grow as slow as the ‘volume’ $|\partial K_{\lambda}|$ of the boundary $\partial K_{\lambda}$, or even slower than $|\partial K_{\lambda}|$, giving rise to ‘edge effects’ which may affect or dominate the limit distribution of $S^{\mathfrak{X}}_{K_{\lambda}}$; see [16, 33].

Probably, the most studied case of limit theorems for RFs deal with rectangular summation regions, which allows for partial sums and limit RFs, similarly as in the case $\nu=1$. Let $\mathfrak{X}=\{\mathfrak{X}(\boldsymbol{t});\boldsymbol{t}\in\mathbb{Z}^{\nu}\}$ be a stationary random field (RF) on $\mathbb{Z}^{\nu},\,\nu\geq 1,\,\boldsymbol{\gamma}=(\gamma_{1},\cdots,\gamma_{\nu})\in\mathbb{R}^{\nu}_{+}$ be a collection of positive numbers, and

[TABLE]

be a family of $\nu$-dimensional ‘rectangles’ indexed by $\lambda>0$, whose sides grow at generally different rate $O(\lambda^{\gamma_{i}}),i=1,\cdots,\nu$ as $\lambda\to\infty$, and

[TABLE]

be the corresponding partial sums RF. [31, 32, 29, 36] discussed the anisotropic scaling limits for any $\boldsymbol{\gamma}\in\mathbb{R}^{\nu}_{+}$ of some classes of LRD RFs $\mathcal{Y}$ in dimension $\nu=2,3$, viz.,

[TABLE]

as $\lambda\to\infty$, where $A_{\lambda,\boldsymbol{\gamma}}\to\infty$ is a normalization. Following [28, 36] the family $\{V^{\mathfrak{X}}_{\boldsymbol{\gamma}};\boldsymbol{\gamma}\in\mathbb{R}^{\nu}_{+}\}$ of all scaling limits in (1.3) will be called the scaling diagram of RF $\mathfrak{X}$. [28] noted that the scaling diagram provides a more complete ‘large-scale summary’ of RF $\mathfrak{X}$ compared to (usual) isotropic or anisotropic scaling at fixed $\boldsymbol{\gamma}\in\mathbb{R}^{\nu}_{+}$ as discussed in [2, 7, 16, 17, 19, 35] and other works.

[31, 32, 29] observed that for a large class of LRD RFs $\mathfrak{X}$ in dimension $\nu=2$, the scaling diagram essentially consists of three elements: $V^{\mathfrak{X}}=\{V^{\mathfrak{X}}_{0},V^{\mathfrak{X}}_{+},V^{\mathfrak{X}}_{-}\}$, $V^{\mathfrak{X}}_{0}$ termed the well-balanced and $V^{\mathfrak{X}}_{\pm}$ the unbalanced scaling limits of $\mathfrak{X}$. For $\nu=2$ without loss of generality (w.l.g.) we can assume $\boldsymbol{\gamma}=(1,\gamma)\in\mathbb{R}^{2}_{+}$ or $\gamma_{1}=1,\gamma_{2}=\gamma$ in (1.2) and denote

[TABLE]

and $V^{\mathfrak{X}}_{\gamma},A_{\lambda,\gamma}$ the corresponding sum in (1.2) and quantities in (1.3) defined for $\gamma>0$, where $K_{[\lambda x,\lambda^{\gamma}y]}=\{(t,s)\in\mathbb{Z}^{2}:1\leq t\leq\lfloor\lambda x\rfloor,1\leq s\leq\lfloor\lambda^{\gamma}y\rfloor\}$. [31, 32, 29] proved that there exists a (nonrandom) $\gamma^{\mathfrak{X}}_{0}>0$ such that $V^{\mathfrak{X}}_{\gamma}$ do not depend on $\gamma$ for $\gamma>\gamma^{\mathfrak{X}}_{0}$ and $\gamma<\gamma^{\mathfrak{X}}_{0}$, viz.,

[TABLE]

and V^{\mathfrak{X}}_{+}\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\neq}}\\ \end{array}aV^{\mathfrak{X}}_{-}\ (\forall\,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, 23, 26, 27, 28, 18, 25]. 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 $\mathbb{Z}^{2}$ having a moving-average representation

[TABLE]

in standardized i.i.d. sequence $\{\varepsilon(u,v);(u,v)\in\mathbb{Z}^{2}\},\mathrm{E}\varepsilon(u,v)=0,\mathrm{E}\varepsilon^{2}(u,v)=1$ with deterministic moving-average coefficients

[TABLE]

$(t,s)\neq(0,0),$ where $q_{i}>0,i=1,2$ satisfy

[TABLE]

which imply $q_{i}>1/2,i=1,2$. In (1.7), $L_{0}(u),u\in[-1,1]$ 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 $\sum_{(t,s)\in\mathbb{Z}^{2}}|a(t,s)|<\infty$ and the ND property of $\mathfrak{X}$ in (1.6) is a consequence of the zero-sum condition:

[TABLE]

In contrast, [29] assumes $1<Q<2$ implying $\sum_{(t,s)\in\mathbb{Z}^{2}}|a(t,s)|=\infty$ and the LRD property of the corresponding linear RF $\mathfrak{X}$ 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 $R_{11},\cdots,R_{33}$ of the parameter set $\{(1/q_{1},1/q_{2}):0<Q<2\}$ of the linear RF $\mathfrak{X}$ in (1.6)-(1.9) with different unbalanced limits. We remark that when $\sum_{(t,s)\in\mathbb{Z}^{2}}a(t,s)\neq 0$ and $Q<1$ the linear RF $\mathfrak{X}$ in (1.6)-(1.7) is SRD and all scaling limits $V^{\mathfrak{X}}_{\gamma},\gamma>0$ agree with Brownian sheet $B_{1/2,1/2}$, see ([29], Theorem 3.4), meaning that in this case the scaling transition does not occur.

Figure 1.

Regions in the parameter set $0<Q<2$ with different unbalanced limits of RF $\mathfrak{X}$ in (1.6)-(1.9). The dashed segment separates the LRD and ND regions.

The regions $R_{11},\cdots,R_{33}$ in Figure 1 are described in Table 1. Recall the definition of fractional Brownian sheet (FBS) $B_{H_{1},H_{2}}=\{B_{H_{1},H_{2}}(x,y);(x,y)\in\mathbb{R}^{2}_{+}\}$ with Hurst parameters $0<H_{1},H_{2}\leq 1$ as a Gaussian process with zero mean and covariance

[TABLE]

see [3]. FBS $B_{H_{1},H_{2}}$ with one of the parameters $H_{i}$ equal to $1/2$ 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 parameter values $(H_{1},H_{1})\in[0,1]^{2}$ by setting

[TABLE]

for any $(x_{i},y_{i})\in\mathbb{R}^{2}_{+},i=1,2$. The limit covariances on the r.h.s. of (1.11) are explicitly written in Remark 2.1. FBS $B_{H_{1},H_{2}}$ where one or both parameters are equal to [math] 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 $\tilde{H}_{i},H_{i},i=1,2$ in Table 1 (expressed in terms of $q_{1},q_{2}$) are specified in the beginning of Sec. 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 $A_{\lambda,\gamma}$ are presented in Sec. 2. The limit distributions in regions $R_{11},R_{12},R_{21},R^{+}_{22}$ refer to LRD set-up and are part of [29]. The new results under ND refer to regions $R_{22}^{-},R_{23},R_{32},R_{33}$ of Table 1. Note $2q_{1}(1-Q)>q_{1}/q_{2}$ in $R_{23}$ and $1/(2q_{2}(1-Q))<q_{1}/q_{2}$ in $R_{32}$. The limit $B_{1/2,0}$ in region $R_{23}$ can be related to the ‘horizontal edge effect’ which dominates the limit distribution of $S^{\mathfrak{X}}_{\lambda,\gamma}$ of (1.4) unless the vertical length $O(\lambda^{\gamma})$ of $K_{[\lambda x,\lambda^{\gamma}y]}$ grows fast enough vs. its horizontal length $O(\lambda)$, or $\gamma>2q_{1}(1-Q)$ holds, in which case FBS $B_{H_{1},1/2}$ dominates. Similarly, $B_{0,1/2}$ in region $R_{32}$ can be related to the ‘vertical edge effect’ appearing in the limit of $S^{\mathfrak{X}}_{\lambda,\gamma}$ unless the vertical length $O(\lambda^{\gamma})$ increases sufficiently slow w.r.t the horizontal length $O(\lambda)$, or $\gamma<1/2q_{2}(1-Q)$ holds, in which case FBS $B_{1/2,H_{2}}$ dominates. Finally, $R_{33}$ can be characterized as the parameter region where the edge effects (either horizontal, or vertical) completely dominate the limit behavior of $S^{\mathfrak{X}}_{\lambda,\gamma}$. The above interpretation of $R_{32},R_{23}$ and $R_{33}$ is based on the approximations of $S^{\mathfrak{X}}_{\lambda,\gamma}$ by suitable ‘edge terms’ which are discussed in Sec. 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., $\gamma_{1}=\cdots=\gamma_{\nu}=1$) star-like regions $K_{\lambda}$ of very general form. This generality of $K_{\lambda}$ 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] consider 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 $q_{1}=q_{2}$. As explained in Remark 2.3, in the case $q_{1}=q_{2}$ and $\boldsymbol{\gamma}=(1,1)$ 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 $\boldsymbol{\gamma}$ and the scaling transition does not exist. See also ([36], Remark 4.1).

The rest of the paper is organized as follows. Sec. 2 contains the main results (Theorems 2.1 - 2.3). The proofs of these results are given in Sec. 3. Sec. 4 presents two examples of fractionally integrated ND RFs, extending the examples of fractionally integrated LRD RFs in [29]. Sec. 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 \begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\longrightarrow}}\\ \end{array},\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{=}}\\ \end{array}, and \begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\neq}}\\ \end{array} for the weak convergence, equality and inequality of finite-dimensional distributions, respectively. $\mathbb{R}^{\nu}_{+}:=\{\boldsymbol{x}=(x_{1},\cdots,x_{\nu})\in\mathbb{R}^{\nu}:x_{i}>0,i=1,\cdots,\nu\},\mathbb{R}_{+}:=\mathbb{R}^{1}_{+}$, $\lfloor x\rfloor:=\max\{k\in\mathbb{Z}:k\leq x\},\lceil x\rceil:=\min\{k\in\mathbb{Z}:k\geq x\},x\in\mathbb{R}$. ${\bf 1}(A)$ stands for the indicator function of a set $A$.

2 Main results

Throughout the paper we use the following notation:

[TABLE]

We recall that $q_{i},i=1,2$ are the asymptotic parameters of the MA coefficients in (1.7) satisfying (1.9) and $Q$ is defined in (1.8). $H_{i},\tilde{H}_{i},i=1,2$ in (2.1) are the Hurst parameters of the limit FBS in Table 1 and in the subsequent Theorems 2.2–2.4. The angular function $L_{0}$ in (1.7) enters the expression of the kernel $a_{\infty}$ in (2.2) through which these limit FBS are defined. The positive quantities $\gamma^{0},\gamma^{0}_{{\rm edge},i},i=1,2$ correspond to the critical points $\gamma^{\mathfrak{X}}_{0}$ in (1.5). Also note that the partition in Figure 1 is formed by segments belonging to the lines $Q_{{\rm edge},i}=1,\widetilde{Q}_{i}=1,i=1,2$ in (2.1) and $Q=1,Q=2$. Let

[TABLE]

Note $a_{\infty}$ is the scaling limit of $a(t,s)$ in (1.7): $\lim_{\lambda\to\infty}\lambda^{q_{1}}a(\lceil\lambda t\rceil,\lceil\lambda^{q_{1}/q_{2}}s\rceil)$ $=a_{\infty}(t,s)$ for any $(t,s)\in\mathbb{R}^{2},(t,s)\neq(0,0).$ We also use the following notation for kernels of the limit RFs expressed as stochastic integrals w.r.t. Gaussian white noise $W(\mathrm{d}u,\mathrm{d}v),(u,v)\in\mathbb{R}^{2},\mathrm{E}W(\mathrm{d}u,\mathrm{d}v)=0,\mathrm{E}(W(\mathrm{d}u,\mathrm{d}v))^{2}=\mathrm{d}u\mathrm{d}v$ on $\mathbb{R}^{2}$. Namely, for $(u,v)\in\mathbb{R}^{2},(x,y)\in\mathbb{R}^{2}_{+}$, let

[TABLE]

We point out that definitions (2.3) apply under ND condition $Q<1$ only. Under the LRD condition $1<Q<2$ the corresponding kernels take a somewhat simpler form:

[TABLE]

(2.4) hold for any $(u,v)\in\mathbb{R}^{2}$. In (2) the kernels $h_{i},i=1,2$ of (2.3) are explicitly written. For the sake of completeness, we also introduce the kernels

[TABLE]

used in the definition of the unbalanced limits in regions $R_{11},R_{12},R_{21}$ of Table 1. Using the above notation we define the well-balanced limit

[TABLE]

in both cases $0<Q<1$ (ND) and $1<Q<2$ (LRD), with respective $h_{0}$ (2.3) and (2.4). Similarly, we define RFs

[TABLE]

and

[TABLE]

Finally, we define RFs

[TABLE]

Recall that the stochastic integral $I(h):=\int_{\mathbb{R}^{2}}h(u,v)W(\mathrm{d}u,\mathrm{d}v)$ w.r.t. the white noise $W$ is well-defined for any $h\in L^{2}(\mathbb{R}^{2})$ and has a Gaussian distribution with zero mean and variance $\mathrm{E}I(h)^{2}=\|h\|^{2}:=\int_{\mathbb{R}^{2}}h(u,v)^{2}\mathrm{d}u\mathrm{d}v$.

Proposition 2.1

The Gaussian RFs $V^{\mathfrak{X}}_{0}=\{V^{\mathfrak{X}}_{0}(x,y);(x,y)\in\mathbb{R}^{2}_{+}\},V^{\mathfrak{X}}_{\pm}=\{V^{\mathfrak{X}}_{\pm}(x,y);(x,y)\in\mathbb{R}^{2}_{+}\}$ are well-defined in respective regions of parameters $q_{i}>0,i=1,2$ as indicated in (2.6), (2.9), (2.12) and (2.13). Moreover,

[TABLE]

and

[TABLE]

where $\sigma_{i}:=\|h_{i}(1,1;\cdot,\cdot)\|<\infty,\tilde{\sigma}_{i}:=\|\tilde{h}_{i}(1,1;\cdot,\cdot)\|<\infty,i=1,2.$

We note that for $1<Q<2$ the statement of Proposition 2.1 is part of ([29], Theorems 3.1-3.3). For $Q<1$ and $h_{0}$ in (2.3) the fact that $h_{0}\in L^{2}(\mathbb{R}^{2})$ (i.e., that $V^{\mathfrak{X}}_{0}$ is well-defined) follows from Lemma 3.1, (3.22)-(3.23). The statements (2.15) and (2.17) for $Q<1$ follow by rewriting the kernels $h_{i},i=1,2$ in (2.3) as

[TABLE]

where

[TABLE]

(note $L_{2,+}=L_{2,-}$ since $a_{\infty}(t,s)=a_{\infty}(t,-s)$ is symmetric in $s$). Furthermore, the above integrals can be rewritten as

[TABLE]

with $H_{i},i=1,2$ defined in (2.1). Whence we immediately see that $h_{i},i=1,2$ factorize into a product of kernels of (fractional) Brownian motions with one-dimensional time, see [34], implying facts (2.15) and (2.17) for $Q<1$.

The main object of this paper is establishing the scaling transition for ND RF $\mathfrak{X}$, viz.,

[TABLE]

where $A_{\lambda,\gamma}\to\infty\,(\lambda\to\infty)$ is normalization and $S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)$ is defined in (1.4).

This question is treated in Theorems 2.2-2.4. In these theorems, $\mathfrak{X}$ is a linear RF as in (1.6) with moving-average coefficients $a(t,s)$ satisfying (1.7), (1.9), where the parameters $q_{1},q_{2}$ satisfy $0<Q<1$ and belong to different regions of Figure 1.

Theorem 2.2

[Region $R^{-}_{22}$: ‘no edge effects’.]* Let $Q_{\rm edge,1}\wedge Q_{\rm edge,2}>1.$ Then the convergence in (2.19) holds with $\gamma^{\mathfrak{X}}_{0}=\gamma^{0}=q_{1}/q_{2},A_{\lambda,\gamma}=\lambda^{H(\gamma)}$ and the limit RFs specified in (2.9), (2.12), viz.,*

[TABLE]

where $H_{i},i=1,2$ and $\sigma_{i},i=1,2$ defined in (2.1) and in Proposition 2.1, respectively.

Define

[TABLE]

The convergence of the above series in the corresponding regions of parameters $q_{1},q_{2}$ will be established later.

Theorem 2.3

[Regions $R_{23}$ and $R_{32}$: ‘one-sided edge effects’.]**

(i) Let $Q_{\rm edge,2}<1<Q_{\rm edge,1}$. Then the convergence in (2.19) for $\gamma\neq\gamma^{\mathfrak{X}}_{0}$ holds with $\gamma^{\mathfrak{X}}_{0}=\gamma^{0}_{\rm edge,2},A_{\lambda,\gamma}=\lambda^{H(\gamma)}$ and the limit RFs specified in (2.9), (2.12), viz.,

[TABLE]

with $H_{1},\sigma_{1}$ as in Theorem 2.2 and $\sigma_{\rm edge,1}<\infty$ defined in (2.20).

(ii) Let $Q_{\rm edge,1}<1<Q_{\rm edge,2}$. Then the convergence in (2.19) for $\gamma\neq\gamma^{\mathfrak{X}}_{0}$ holds with $\gamma^{\mathfrak{X}}_{0}=\gamma^{0}_{\rm edge,1},A_{\lambda,\gamma}=\lambda^{H(\gamma)}$ and the limit RFs specified in (2.9), (2.12), viz.,

[TABLE]

with $H_{2},\sigma_{2}$ as in Theorem 2.2 and $\sigma_{\rm edge,2}<\infty$ defined in (2.20).

Theorem 2.4

[Region $R_{33}$: ‘two-sided edge effects’.]* Let $Q_{\rm edge,1}\vee Q_{\rm edge,2}<1$. Then the convergence in (2.19) holds with $\gamma^{\mathfrak{X}}_{0}=1,A_{\lambda,\gamma}=\lambda^{H(\gamma)}$ and the limit RFs specified in (2.9), (2.12), viz.,*

[TABLE]

where $B_{1/2,0}$ and $B_{0,1/2}$ are independent RFs and $\sigma_{\rm edge,i},i=1,2$ are given in (2.20).

Remark 2.1

From (1.11) for any $(x_{i},y_{i}),i=1,2$ we get that

[TABLE]

Particularly,

[TABLE]

The FBS $B_{H_{1},H_{2}}$ with $H_{1}\cdot H_{2}=0$ can be defined by Kolmogorov’s theorem as Gaussian probability measures on the space $\mathbb{R}^{\mathbb{R}^{2}_{+}}$ equipped with the $\sigma$-field generated by cylinder sets. They have very irregular (nonseparable) paths due to the fact that the covariance functions in (2.21) are discontinuous at $(x_{1},y_{1})=(x_{2},y_{2})$. Finite dimensional distributions of these RFs can be easily identified using independent FBM. For instance, the (joint) distribution of $B_{1/2,0}(x_{i},y_{i}),(x_{i},y_{i})\in\mathbb{R}^{2}_{+},i=1,\cdots,m,0<y_{1}<\cdots<y_{m}$ coincides with that of the random vector

[TABLE]

where $B^{(i)}=\{B^{(i)}(x);x>0\},i=0,1,\cdots,m$ are independent standard Brownian motions. Note for each $x>0$ fixed, \{B_{1/2,0}(x,y);y>0\}\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{=}}\\ \end{array}\{W(0)-W(y);y>0\} where $W(y)\sim N(0,x/2)\,(\forall\,y\geq 0)$ and $\mathrm{E}W(y)W(y^{\prime})=0\ (y\neq y^{\prime},y,y^{\prime}\geq 0)$. We note that the above extension of FBS satisfy the basic (multi) self-similarity property (c.f. [10]): for any $(H_{1},H_{2})\in[0,1]^{2},\lambda_{1}>0,\lambda_{2}>0$

[TABLE]

Remark 2.2

The ‘edge effects’ in (i) and (ii) of Theorem 2.3 may be also respectively labelled as ‘horizontal’ and ‘vertical’. Note the normalizations $\lambda^{H(\gamma)}=\lambda^{1/2}$ in (i), $\gamma<\gamma^{0}_{\rm edge,2}$ (respectively, $\lambda^{H(\gamma)}=\lambda^{\gamma/2}$ in (ii), for $\gamma>\gamma^{0}_{\rm edge,1}$) is proportional to the the square root of the horizontal (respectively, vertical) length of the rectangle $K_{[\lambda x,\lambda^{\gamma}y]}$ and $S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)$ behaves as a sum of weakly dependent r.v.’s indexed by points on the horizontal (respectively, vertical) edges of this rectangle. See Remark 3.1 below.

Remark 2.3

In the ‘isotropic’ case $q_{1}=q_{2}=:\beta$ we have that $Q=2/\beta\in(0,1)$ is equivalent to $\beta\in(2,\infty)$ and $Q_{{\rm edge},1}=Q_{{\rm edge},2}=5/(2\beta)$. In this case either Theorem 2.2 (for $2<\beta<5/2$) or Theorem 2.4 (for $5/2<\beta<\infty$) applies and our results can be related to ([16], Theorems 4.1 and 4.3). In the case $2<\beta<5/2$ Theorem 2.2 for $\gamma=\gamma^{0}=1$ yields the Gaussian limit with the variance $\|h_{0}(x,y;\cdot,\cdot)\|^{2}$ which coincides with the variance in ([16], (4.2)). In the interval $5/2<\beta<\infty$ the coincidence of the limiting variances in Theorem 2.2, $\gamma=1$ and ([16], (4.5)) is less straightforward and can be verified by noting that for any $\delta>0$

[TABLE]

where $\partial K_{[\lambda x,\lambda y]}=\{(t,s)\in K_{[\lambda x,\lambda y]}:{\rm dist}((t,s),K^{c}_{[\lambda x,\lambda y]})=1\}$ is the boundary of $K_{[\lambda x,\lambda y]},K^{c}_{[\lambda x,\lambda y]}=\mathbb{Z}^{2}\setminus K_{[\lambda x,\lambda y]}$ and ${\rm dist}((u,v),A)=\inf_{(t,s)\in A}\|(t,s)-(u,v)\|$ is the distance between $(u,v)\in\mathbb{Z}^{2}$ and $A\subset\mathbb{Z}^{2}$. (2.24) follows by using the summability properties of $|a(t,s)|$ guaranteeing the convergence of the series in (2.20) and the zero-sum condition (1.9).

Remark 2.4

In Theorem 2.3 the convergence in (2.19) at $\gamma=\gamma^{\mathfrak{X}}_{0}$ is an open question. We conjecture that the corresponding limits agree with $B_{1/2,0}$ for $\gamma^{\mathfrak{X}}_{0}=\gamma^{0}_{\rm edge,2}$ and $B_{0,1/2}$ for $\gamma^{\mathfrak{X}}_{0}=\gamma^{0}_{\rm edge,1}$ however the normalizations might include additional logarithmic factors making the proofs more difficult. A similar difficulty may accompany the attempts to extend (2.19) to values $q_{1},q_{2}$ belonging to the boundaries of the partition in Figure 1, particularly to the segments $\widetilde{Q}_{i}=1,Q_{\rm edge,i}=1,i=1,2$. On the other hand, the normalizing exponent $H(\gamma)\equiv H(\gamma,q_{1},q_{2})$ in Theorems 2.2-2.4 and in ([29], Theorems 3.1-3.3, case $k=1$) extends by continuity to a (jointly) continuous function on the set $\{\gamma>0,0<Q<2\}$ suggesting that the above extension of the scaling transition is plausible.

Remark 2.5

Note $\sigma_{\rm edge,1}$ and/or $\sigma_{\rm edge,2}$ in (2.20) may vanish provided

[TABLE]

hold. Clearly, each of the conditions in (2.25) imply (1.9) and the ND property of the corresponding RF, but the converse implication is not true. Relations (2.25) may be termed the *vertical * (respectively, horizontal) ND property. Under (2.25) the corresponding limits in Theorems 2.3 and 2.4 are trivial, while nontrivial limits may exists under different conditions on $\gamma$ and $q_{1},q_{2}$, eventually changing Figure 1 and Table 1.

3 Proofs of Theorems 2.2-2.4

3.1 Outline of the proof and preliminaries

Let us explain the main steps of the proof of Theorems 2.2-2.4. By definition, $S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)$ can be rewritten as a weighted sum

[TABLE]

in i.i.d.r.v.s $\varepsilon(u,v),(u,v)\in\mathbb{Z}^{2}$. It happens that different regions of ‘noise locations’ $(u,v)$ contribute to different limit distributions in our theorems.

Figure 2.

Partition of $\mathbb{Z}^{2}$ by $K=K_{[\lambda x,\lambda^{\gamma}y]}$ and 8 ‘outer’ sets $K^{c}_{i,j},i,j=1,0,-1,(i,j)\neq(0,0)$. The shaded regions $\partial_{0,1}K,\partial_{0,-1}K$ (respectively, $\partial_{1,0}K,\partial_{-1,0}K$) are related to the ‘horizontal (respectively, vertical) edge effect’.

The basic decomposition

[TABLE]

of $\mathbb{Z}^{2}$ into 9 sets is shown in Figure 2. Following (3.2) we decompose (3.1) as

[TABLE]

where, with zero-sum condition (1.9) in mind,

[TABLE]

As shown in Lemma 3.2,

• •

terms $S(K),S(K^{c}_{-1,0}),S(K^{c}_{1,0})$ in (3.3) contribute to the FBS limit $\sigma_{1}B_{H_{1},1/2}$ for $\gamma>\gamma^{\mathfrak{X}}_{0}$ and the remaining terms in (3.3) are negligible;

• •

terms $S(K),S(K^{c}_{0,-1}),S(K^{c}_{0,1})$ in (3.3) contribute to the FBS limit $\sigma_{2}B_{1/2,H_{2}}$ for $\gamma<\gamma^{\mathfrak{X}}_{0}$ and the remaining terms in (3.3) are negligible;

• •

all terms on the r.h.s. of (3.3) are relevant for proving the well-balanced limit $V^{\mathfrak{X}}_{0}$ in Theorem 2.2;

• •

the main contribution to the unbalanced limit $\sigma_{\rm edge,1}B_{1/2,0}$ in Theorems 2.3-2.4 comes from innovations ‘close’ to the horizontal edges of $K$ (the shaded regions $\partial_{0,1}K$ and $\partial_{0,-1}K$ in Figure 2);

• •

the main contribution to the unbalanced limit $\sigma_{\rm edge,2}B_{0,1/2}$ in Theorems 2.3-2.4 comes from innovations ‘close’ to the vertical edges of $K$ (the shaded regions $\partial_{1,0}K$ and $\partial_{-1,0}K$ in Figure 2).

After identification of the main terms, the limit distributions of these terms is obtained using a general criterion for the convergence of linear forms in i.i.d.r.v.s towards stochastic integral w.r.t. white noise. Consider a linear form

[TABLE]

with real coefficients $\sum_{(u,v)\in\mathbb{Z}^{2}}g(u,v)^{2}<\infty$. The following proposition is a version of ([12], Proposition 14.3.2) and ([36], Proposition 5.1) and we omit the proof.

Proposition 3.1

Let $S(g_{\lambda}),\lambda>0$ be as in (3.5). Suppose $g_{\lambda}(u,v)$ are such that for a real-valued function $h\in L^{2}(\mathbb{R}^{2})$ and some $m_{i}=m_{i}(\lambda)\to\infty,\lambda\to\infty,i=1,2$ the functions

[TABLE]

tend to $h$ in $L^{2}(\mathbb{R}^{2})$, viz.,

[TABLE]

Then

[TABLE]

3.2 Two auxiliary lemmas

The proofs of our results requires evaluation of various multiple series involving the coefficients $a(t,s)$. Similarly as in [29, 36] these series are estimated by corresponding multiple integrals involving the function $a_{\infty}(t,s)$ in (2.2) or

[TABLE]

Indeed, by elementary inequalities

[TABLE]

see ([36], (2.16)) the function $\rho$ in (3.11)-(3.26) can be replaced by $|a_{\infty}|$ while most multiple sums involving $a(t,s)$ in the proofs below can be evaluated by corresponding multiple integrals in Lemma 3.1.

Lemma 3.1

Let $\rho(t,s)$ be defined as in (3.9), where $q_{i}>0,i=1,2$. Then for any $\delta>0,h>0$

[TABLE]

In addition, let $Q<1$. Then

[TABLE]

[TABLE]

Furthermore, as $\mu\to\infty$

[TABLE]

Proof. Relations (3.11) and (3.12) are proved in ([29], Proposition 5.1).

Proof of (3.13). Write the l.h.s. of (3.13) as I=\int_{0}^{\infty}\mathrm{d}u\big{(}\int_{1}^{\infty}\mathrm{d}t\int_{0}^{\infty}(|t+u|^{q_{1}}+|s|^{q_{2}})^{-1}\mathrm{d}s\big{)}^{2}=C\int_{0}^{\infty}\mathrm{d}u\big{(}\int_{1}^{\infty}\mathrm{d}t(t+u)^{\gamma^{0}-q_{1}}\big{)}^{2}=\int_{0}^{1}\mathrm{d}u\big{(}\int_{1}^{\infty}t^{\gamma^{0}-q_{1}}\mathrm{d}t\big{)}^{2}+\int_{1}^{\infty}\mathrm{d}u(u^{\gamma^{0}-q_{1}+1})^{2}<\infty since $2(q_{1}-1-\gamma^{0})>1$. The proof of (3.15) is analogous.

Proof of (3.16). By integrating the inner integral w.r.t. $t\in(0,\infty)$ the l.h.s. of (3.16) can be written as I=C\int_{0}^{\infty}\mathrm{d}v\big{(}\int_{0}^{1}(s+v)^{(q_{2}/q_{1})-q_{2}}\,\mathrm{d}s\big{)}^{2}=\int_{0}^{1}\mathrm{d}v(\cdots)^{2}+\int_{1}^{\infty}\mathrm{d}v(\cdots)^{2}=:I_{1}+I_{2} where $I_{1}\leq C\int_{0}^{1}v^{2(1+(q_{2}/q_{1})-q_{2})}\mathrm{d}v<\infty$ since $Q_{\rm edge,2}>1$ and $I_{2}\leq C\int_{1}^{\infty}v^{2((q_{2}/q_{1})-q_{2})}\mathrm{d}v<\infty$ since $(1/q_{1})+(1/2q_{2})<1$. The proof of (3.14) is analogous.

Proof of (3.17). Write the l.h.s. of (3.17) as $I=I_{1}+I_{2}$ where $I_{1}:=\int_{1}^{\infty}\mathrm{d}u\cdots,I_{2}:=\int_{0}^{1}\mathrm{d}u\cdots$. Then $I_{1}\leq C\int_{1}^{\infty}\rho(u,0)^{2}\mathrm{d}u<\infty$ and, by change of variables $t\to ut,s\to u^{q_{1}/q_{2}}s$, $I_{2}\leq\int_{0}^{1}u^{2+2(q_{1}/q_{2})-2q_{1}}\mathrm{d}u<C<\infty$ since $\int_{0}^{1/u}\mathrm{d}t\int_{0}^{1/u^{q_{1}/q_{2}}}\rho(t+1,s-(v/u^{q_{1}/q_{2}}))\mathrm{d}s$ $\leq C\int_{0}^{\infty}\mathrm{d}t\int_{\mathbb{R}}\rho(t+1,s)\,\mathrm{d}s<\infty$ in view of (3.15). The proof of (3.18) is analogous.

Proof of (3.19). Split the l.h.s. of (3.19) as $I=I_{1}+I_{2}$, where $I_{1}:=\int_{(0,\infty)^{2}\setminus(0,1)^{2}}\mathrm{d}u\mathrm{d}v\cdots,I_{2}:=\int_{(0,1)^{2}}\mathrm{d}u\mathrm{d}v\cdots$. Then $I_{1}\leq C\int_{(0,\infty)^{2}\setminus(0,1)^{2}}\rho(u,v)^{2}\mathrm{d}u\mathrm{d}v\leq C$ due to (3.12). Next by change of variables: $t\to\rho(u,v)^{-1/q_{1}}t,s\to\rho(u,v)^{-1/q_{2}}s$ and using $\rho(u+t\rho(u,v)^{-1/q_{1}},v+s\rho(u,v)^{-1/q_{2}})=\rho(u,v)\rho(t+u\rho(u,v)^{1/q_{1}},s+v\rho(u,v)^{1/q_{2}})\ \leq\rho(u,v)(\rho(1,1)\wedge\rho(t,t)),t,s\geq 0$ we obtain

[TABLE]

where the first integral converges due to (3.11) and $Q>2/3$, and the second due to (3.12) and $Q<1$. This proves (3.19).

Proof of (3.20). Split the l.h.s. of (3.20) as $I=I_{1}+I_{2}$, where $I_{1}:=\int_{(0,1)\times\mathbb{R}_{+}}\mathrm{d}u\mathrm{d}v\cdots,I_{2}:=\int_{(1,\infty)\times\mathbb{R}_{+}}\mathrm{d}u\mathrm{d}v\cdots$. Then I_{1}\leq C\int_{0}^{\infty}\mathrm{d}v\big{(}\int_{1}^{\infty}(s+v)^{-q_{2}}\mathrm{d}s\big{)}^{2}\leq C\int_{0}^{\infty}\mathrm{d}v\big{(}1\wedge v^{-2(q_{2}-1)}\big{)}<\infty since $2(q_{2}-1)>1$ follows from $3/q_{2}<2$. Next, I_{2}\leq C\int_{1}^{\infty}\mathrm{d}u\int_{0}^{\infty}\mathrm{d}v\big{(}\int_{1}^{\infty}(u^{q_{1}}+(s+v)^{q_{2}})^{-1}\mathrm{d}s\big{)}^{2}\leq CJ_{1}J_{2}, where $J_{1}:=\int_{1}^{\infty}u^{3(q_{1}/q_{2})-2q_{1}}\mathrm{d}u<\infty$ for $\frac{1}{q_{1}}+\frac{3}{q_{2}}<2$ and

[TABLE]

since $q_{2}>3/2$. This proves (3.20). The proof of (3.21) is analogous.

Proof of (3.22). Split $\mathbb{R}^{2}\setminus[0,1]^{2}=\cup_{i=1}^{8}D_{i}$ similarly as in Fig. 2 and then the l.h.s. of (3.21) as $I=\sum_{i=1}^{8}I_{i}$ where I_{i}:=\int_{D_{i}}\big{(}\int_{[0,1]^{2}}\rho(t-u,s-v)\mathrm{d}t\mathrm{d}s\big{)}^{2}\mathrm{d}u\mathrm{d}v, $D_{1}:=(-\infty,0)^{2},D_{1}:=(-\infty,0)\times(0,1),D_{3}:=(0,1)\times(-\infty,0)$ etc. Here, $I_{1}<\infty$ according to (3.19). Similarly, I_{2}=\int_{(0,\infty)\times(0,1)}\mathrm{d}u\mathrm{d}v\big{(}\int_{(0,1]^{2}}\rho(t+u,s-v)\mathrm{d}t\mathrm{d}s\big{)}^{2}<\infty and I_{3}=\int_{(0,1)\times(0,\infty)}\mathrm{d}u\mathrm{d}v\big{(}\int_{(0,1]^{2}}\rho(t-u,s+v)\mathrm{d}t\mathrm{d}s\big{)}^{2}<\infty follow from (3.17) and (3.18), respectively. The remaining relations $I_{i}<\infty,i=3,\cdots,8$ follow from (3.17)-(3.19) in a similar way, proving (3.22).

Proof of (3.23). Split $\mathbb{R}^{2}\setminus[0,1]^{2}=\cup_{i=0}^{8}D^{\prime}_{i},$ where $D^{\prime}_{0}:=\mathbb{R}^{2}\setminus[-1,2]^{2},D^{\prime}_{1}:=(-1,0)^{2},D^{\prime}_{2}:=(-1,0)\times(0,1),D^{\prime}_{3}:=(-1,0)\times(1,2)$ etc. Accordingly, for the l.h.s. $J$ of (3.23) we have $J\leq C\sum_{i=0}^{8}J^{\prime}_{i}$, where J^{\prime}_{i}:=\int_{[0,1]^{2}}\mathrm{d}u\mathrm{d}v\big{(}\int_{D^{\prime}_{i}}\rho(t-u,s-v)\mathrm{d}t\mathrm{d}s\big{)}^{2},i=0,1,\cdots,8. Note $\rho(t-u,s-v)\leq\delta$ for some $\delta>0$ and all $(s,v)\in[0,1]^{2},(t,s)\in D^{\prime}_{0}$, implying J^{\prime}_{0}\leq\big{(}\int_{\mathbb{R}^{2}}\rho(t,s){\bf 1}(\rho(t,s)\leq\delta)\mathrm{d}t\mathrm{d}s\big{)}^{2}<\infty according to (3.12). The proof of $J^{\prime}_{i}<\infty,i=1,\cdots,8$ uses (3.17)-(3.19) and is analogous to that of $I_{i}<\infty,i=1,\cdots,8$ in the proof of (3.22) above. For instance, J^{\prime}_{2}=\int_{[0,1]^{2}}\mathrm{d}u\mathrm{d}v\big{(}\int_{(-1,0)\times(0,1)}\rho(t-u,s-v)\mathrm{d}t\mathrm{d}s\big{)}^{2}=\int_{[0,1]^{2}}\mathrm{d}u\mathrm{d}v\big{(}\int_{(0,1)^{2}}\rho(t+u,s-v)\mathrm{d}t\mathrm{d}s\big{)}^{2}<\infty according to (3.17). This proves (3.23).

Proof of (3.24). Split the l.h.s. of (3.24) as $J_{\mu}=\int_{\mathbb{R}^{2}_{+}\setminus[0,1]^{2}}\mathrm{d}u\mathrm{d}v\cdots+\int_{[0,1]^{2}}\mathrm{d}u\mathrm{d}v\cdots=:J^{\prime}_{\mu}+J^{\prime\prime}_{\mu}.$ Then J^{\prime\prime}_{\mu}\leq\big{(}\int_{1}^{\infty}\rho(0,s)\mathrm{d}s\big{)}^{2}\leq C since $q_{2}>1$. Next, consider $J^{\prime}_{\mu}=\int_{0}^{1}\mathrm{d}u\int_{1}^{\infty}\mathrm{d}v\cdots+\int_{1}^{\infty}\mathrm{d}u\int_{0}^{\infty}\mathrm{d}v\cdots=:I^{\prime}_{\mu}+I^{\prime\prime}_{\mu}$. We have I^{\prime}_{\mu}\leq\int_{1}^{\infty}\mathrm{d}v\big{(}\int_{1}^{\mu}\rho(0,s+v)\mathrm{d}s\big{)}^{2}=\int_{1}^{\mu}\mathrm{d}v\big{(}\int_{1}^{\mu}(s+v)^{-q_{2}}\mathrm{d}s\big{)}^{2}+\int_{\mu}^{\infty}(\mu/v^{q_{2}})^{2}\mathrm{d}v and hence

[TABLE]

satisfies the bound in (3.24).

Consider $I^{\prime\prime}_{\mu}=\int_{1}^{\infty}\mathrm{d}u\int_{0}^{\mu}\mathrm{d}v\cdots+\int_{1}^{\infty}\mathrm{d}u\int_{\mu}^{\infty}\mathrm{d}v\cdots=:L^{\prime}_{\mu}+L^{\prime\prime}_{\mu}$. Here, L^{\prime\prime}_{\mu}\leq\int_{1}^{\infty}\mathrm{d}u\int_{\mu}^{\infty}(\mu\rho(u,v)\big{)}^{2}\mathrm{d}v=\mu^{2}\int_{1}^{\infty}\mathrm{d}u $\int_{\mu}^{\infty}(u^{q_{1}}+v^{q_{2}})^{-2}\mathrm{d}v=\mu^{2}\int_{1}^{\infty}u^{(q_{1}/q_{2})-2q_{1}}\mathrm{d}u\int_{\mu/u^{q_{1}/q_{2}}}^{\infty}(1+v^{q_{2}})^{-2}\mathrm{d}v$ where the last integral is evaluated by splitting it over $u<\mu^{q_{2}/q_{1}}$ and $u>\mu^{q_{2}/q_{1}}$ yielding

[TABLE]

Finally, $L^{\prime}_{\mu}=\int_{1}^{\mu^{q_{2}/q_{1}}}\mathrm{d}u\int_{0}^{\mu}\mathrm{d}v\cdots+\int_{\mu^{q_{2}/q_{1}}}^{\infty}\mathrm{d}u\int_{0}^{\mu}\mathrm{d}v\cdots=:L^{\prime}_{\mu,1}+L^{\prime}_{\mu,2}$ where

[TABLE]

and L^{\prime}_{\mu,2}\leq C\mu^{3}\int_{\mu^{q_{2}/q_{1}}}^{\infty}u^{-2q_{1}}\mathrm{d}u\ =\ O\big{(}\mu^{3-2q_{2}+(q_{2}/q_{1})}\big{)}, proving (3.24). The proof of (3.25) is completely analogous, by exchanging $t$ and $s$.

Proof of (3.26). Split the l.h.s. of (3.26) as $I_{\mu}=\int_{0}^{1}\mathrm{d}v(\cdots)^{2}+\int_{1}^{\infty}\mathrm{d}v(\cdots)^{2}=:I_{\mu,1}+I_{2,\mu}$. Then I_{1,\mu}\leq\big{(}\int_{1}^{\infty}\rho(0,s)\mathrm{d}s\big{)}^{2}<C and I_{2,\mu}\leq\int_{1}^{\infty}\mathrm{d}v\big{(}\int_{1}^{\mu}(s+v)^{-q_{2}}\mathrm{d}s\big{)}^{2} satisfies the bound in (3.26) since $q_{2}>1$. This proves (3.26) and completes the proof of the lemma. $\Box$

We use the above lemma for evaluation of ‘remainder terms’ in (3.4), where

[TABLE]

the remaining sets analogously defined, viz., $K^{c}_{-1,0}:=\{(u,v)\in\mathbb{Z}^{2}:u\leq 0,1\leq v\leq[\lambda^{\gamma}y]\},$ $K^{c}_{1,0}:=\{(u,v)\in\mathbb{Z}^{2}:u>[\lambda x],1\leq v\leq[\lambda^{\gamma}y]\},$ $K^{c}_{-1,1}:=\{(u,v)\in\mathbb{Z}^{2}:u\leq 0,v>[\lambda^{\gamma}y]\},$ $K^{c}_{0,1}:=\{(u,v)\in\mathbb{Z}^{2}:1\leq u\leq[\lambda x],v>[\lambda^{\gamma}y]\},$ $K^{c}_{1,1}:=\{(u,v)\in\mathbb{Z}^{2}:u>[\lambda x],v>[\lambda^{\gamma}y]\}.$

Lemma 3.2

Let $0<Q<1,Q_{\rm edge,i}\neq 1,i=1,2$. Then

[TABLE]

in the following cases:

[TABLE]

and

[TABLE]

Proof. In view of the bound in (3.10) and the reflection symmetry of the function $\rho(t,s)$ w.r.t. the coordinate axes, it suffices to prove (3.29) for $(i,j)=(-1,-1)$ and $(i,j)=(-1,0)$. Moreover, it suffices to consider $x=y=1$.

Case $(i,j)=(-1,-1)$. Denote $J_{\lambda,\gamma}$ the l.h.s. of (3.28) for $x=y=1,i=j=-1$. Then using (3.10)

[TABLE]

Then from (3.31), (3.21), (3.22), (3.24) with $\mu=\lambda^{\gamma-\gamma_{0}}$ we obtain that

[TABLE]

Then (3.28) follows from

[TABLE]

with $H(\gamma)$ defined in Theorems 2.2-2.4. The latter theorems determine four of cases of $Q_{{\rm edge},i},i=1,2$ in which $\gamma^{\mathfrak{X}}_{0}$ takes different values.

($C$1)

Case $Q_{\rm edge,1}>1$. Then $\gamma^{\mathfrak{X}}_{0}\geq\gamma^{0},2H(\gamma)=3+2\gamma^{0}-2q_{1}+\gamma$ (see Theorems 2.2 and 2.3 (i)) and $\phi_{+}(\gamma)=3+3\gamma^{0}-2q_{1}+(\gamma-\gamma_{0})(3+(1/\gamma^{0})-2q_{2})_{+}<3+2\gamma^{0}-2q_{1}+\gamma=2H(\gamma)$ follows by $\gamma>\gamma^{0},Q<1$;

($C$2)

Case $Q_{\rm edge,1}<1<Q_{\rm edge,2}$. Then $\gamma^{\mathfrak{X}}_{0}=\gamma^{0}_{\rm edge,1}<\gamma^{0},2H(\gamma)=\gamma$. For $\gamma^{0}_{\rm edge,1}<\gamma<\gamma^{0}$ we have that $((3/\gamma^{0})+1-2q_{2})_{+}=0,\phi_{-}(\gamma)=\gamma((3/\gamma^{0})+3-2q_{2})$ and $\phi_{-}(\gamma)<2H(\gamma)$ reduces to $Q_{\rm edge,1}<1$; for $\gamma\geq\gamma^{0}$ relation $\phi_{+}(\gamma)=3+3\gamma^{0}-2q_{1}+(\gamma-\gamma^{0})(3+(1/\gamma^{0})-2q_{2})_{+}<2H(\gamma)=\gamma$ follows similarly;

($C$3)

Case $Q_{\rm edge,1}\vee Q_{\rm edge,2}<1$. Then $\gamma^{\mathfrak{X}}_{0}=1,2H(\gamma)=\gamma\ (\gamma>\gamma^{\mathfrak{X}}_{0})$ (see Theorem 2.4), $(3+(1/\gamma^{0})-2q_{2})_{+}=0,((3/\gamma^{0})+1-2q_{2})_{+}=0$ and $\phi_{+}(\gamma)=3+3\gamma^{0}-2q_{1},$ $\phi_{-}(\gamma)=\gamma((3/\gamma^{0})+3-2q_{2})$. Then $\phi_{+}(\gamma)<\gamma\,(\gamma\geq\gamma_{0}\vee 1),\,\phi_{-}(\gamma)<\gamma\,(1<\gamma\leq\gamma^{0}\vee 1)$ follow by $Q_{{\rm edge},i}<1,i=1,2$.

Case $(i,j)=(0,-1)$. Write $\widetilde{J}_{\lambda,\gamma}$ for the l.h.s. of (3.28) for the above $i,j$ and $x=y=1$. Then similarly to (3.31)

[TABLE]

where

[TABLE]

follow from (($C$3)), (3.34) and (3.15), (3.16). Then (3.28) for $(i,j)=(0,-1)$ follows from

[TABLE]

with $H(\gamma)$ defined in Theorems 2.2-2.4.

($\widetilde{C}$1)

Case $Q_{\rm edge,1}>1$. Then $\gamma^{\mathfrak{X}}_{0}\geq\gamma^{0},2H(\gamma)=3+2\gamma^{0}-2q_{1}+\gamma$ and $\tilde{\phi}_{+}(\gamma)=3+3\gamma^{0}-2q_{1}+(\gamma-\gamma_{0})(3-2q_{2})_{+}<2H(\gamma)$ as in ($C$1);

($\widetilde{C}$2)

Case $Q_{\rm edge,1}<1<Q_{\rm edge,2}$. Then $\gamma^{\mathfrak{X}}_{0}=\gamma^{0}_{\rm edge,1}<\gamma^{0},2H(\gamma)=\gamma$. For $\gamma^{0}_{\rm edge,1}<\gamma<\gamma^{0}$ we have that $\widetilde{\phi}_{-}(\gamma)=1+(2+3\gamma^{0}-2q_{1})(\gamma/\gamma^{0})<\gamma=2H(\gamma)$ is equivalent to $\gamma>\gamma^{0}_{\rm edge,1}$ (condition $1>(1/q_{1})+(1/2q_{2})$ in (3.35) is satisfied by $Q_{\rm edge,1}<1$); for $\gamma\geq\gamma^{0}$ relation $\widetilde{\phi}_{+}(\gamma)<\gamma$ follows by $Q_{\rm edge,1}<1$;

($\widetilde{C}$3)

Case $Q_{\rm edge,1}\vee Q_{\rm edge,2}<1$. Then $\gamma^{\mathfrak{X}}_{0}=1,2H(\gamma)=\gamma\ (\gamma>\gamma^{\mathfrak{X}}_{0})$. Let $\gamma>\gamma^{0}\vee 1$ then $\widetilde{\phi}_{+}(\gamma)<\gamma$ as in ($\widetilde{C}$2). Next, let $1<\gamma\leq\gamma^{0}$ then $\widetilde{\phi}_{-}(\gamma)=1<\gamma=2H(\gamma)$.

The above discussion proves (3.28) for (3.29). The proof of (3.28) for (3.30) is analogous and omitted. Lemma 3.2 is proved. $\Box$

3.3 Proof of Theorem 2.2

The finite-dimensional convergence in (2.19), or A^{-1}_{\lambda,\gamma}S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\longrightarrow}}\\ \end{array}V^{\mathfrak{X}}_{\gamma}(x,y), towards the corresponding limit RF $V^{\mathfrak{X}}_{\gamma}$ is equivalent to the one-dimensional convergence of arbitrary linear combinations, viz., $A^{-1}_{\lambda,\gamma}\sum_{\ell=1}^{p}\theta_{\ell}S^{\mathfrak{X}}_{\lambda,\gamma}(x_{\ell},y_{\ell})$ \begin{array}[t]{c}\stackrel{{\scriptstyle\rm d}}{{\longrightarrow}}\\ \end{array}\sum_{\ell=1}^{p}V^{\mathfrak{X}}_{\gamma}(x_{\ell},y_{\ell}) for any $\theta_{\ell}\in\mathbb{R},(x_{\ell},y_{\ell})\in\mathbb{R}^{2}_{+},\ell=1,\cdots,p,$ $p\geq 1.$ Towards this aim we use Proposition 3.1 and write $A^{-1}_{\lambda,\gamma}\sum_{\ell=1}^{p}\theta_{\ell}S^{\mathfrak{X}}_{\lambda,\gamma}(x_{\ell},y_{\ell})=Q(g_{\lambda,\gamma})$ as a linear form in (3.5) with coefficients $g_{\lambda,\gamma}(u,v)=\sum_{\ell=1}^{p}\theta_{\ell}g_{\lambda,\gamma}(x_{\ell},y_{\ell};u,v)$ where

[TABLE]

Since in all cases of Theorem 2.2 the limit RF $V^{\mathfrak{X}}_{\gamma}(x,y)=\int_{\mathbb{R}^{2}}h(x,y;u,v)W(\mathrm{d}u,\mathrm{d}v)$ writes as a white noise integral w.r.t. some $h(x,y;\cdot,\cdot)\in L^{2}(\mathbb{R}^{2})$, by Proposition 3.1 it it suffices to prove the $L^{2}$-convergence

[TABLE]

with $h(u,v):=\sum_{\ell=1}^{p}\theta_{\ell}h_{\gamma}(x_{\ell},y_{\ell};u,v)$ and suitably chosen $m_{i}=m_{i}(\lambda),i=1,2$, see (3.6), (3.7). Clearly, the proof of (3.38) can be reduced to the case $p=\theta_{1}=1$ and $(x_{1},y_{1})=(x,y)\in\mathbb{R}^{2}_{+}$ fixed. Accordingly, the proof of (2.19) is reduced to one-dimensional convergence A^{-1}_{\lambda,\gamma}S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)\begin{array}[t]{c}\stackrel{{\scriptstyle\rm d}}{{\longrightarrow}}\\ \end{array}V^{\mathfrak{X}}_{\gamma}(x,y) at a given point $(x,y)\in\mathbb{R}^{2}_{+}$, or the verification of (3.38) for $g_{\lambda,\gamma}(\cdot,\cdot)\equiv g_{\lambda,\gamma}(x,y;\cdot,\cdot)$ in (3.37) and $h(\cdot,\cdot)\equiv h_{j}(x,y;\cdot,\cdot),j=0,1,2$ in (2.4) in respective cases $\gamma=\gamma^{0},\gamma>\gamma^{0},\gamma<\gamma^{0}$. For this, it is convenient to rewrite $g_{\lambda,\gamma}$ and $\tilde{g}_{\lambda,\gamma}$ as integrals of step functions and change the variables $(t,s)\to(\tilde{m}_{1}t,\tilde{m}_{2}s):\mathbb{R}^{2}\to\mathbb{R}^{2}$, with $\tilde{m}_{i},i=1,2$ defined below. Then using (1.9)

[TABLE]

where

[TABLE]

The normalising factor $\kappa_{\lambda,\gamma}$ will be used to prove the limit of the integrand on the r.h.s. of (3.39).

Consider (3.38) for $\gamma=\gamma^{0}$. Let $m_{1}=\tilde{m}_{1}:=\lambda,m_{2}=\tilde{m}_{2}:=\lambda^{\gamma^{0}}.$ Then \widetilde{K}_{\lambda,\gamma^{0}}(x,y)=\big{(}0,\lfloor\lambda x\rfloor/\lambda\big{]}\times\big{(}0,\lfloor\lambda^{\gamma^{0}}y\rfloor/\lambda^{\gamma^{0}}\big{]},\kappa_{\lambda,\gamma^{0}}=\lambda^{q_{1}} and

[TABLE]

where

[TABLE]

for any $(t,s)\neq(s,v)$. Moreover,

[TABLE]

see ([29], (7.28)), with $C>0$ independent of $\lambda\geq 1$ and $\rho$ as in (3.9). Whence, the point-wise convergence $\tilde{g}_{\lambda,\gamma_{0}}(x,y;u,v)\to h_{0}(x,y;u,v),(u,v)\in\mathbb{R}^{2},(u,v)\not\in\partial K(x,y)$ (= the boundary of $K(x,y):=(0,x]\times(0,y]$) easily follows. Moreover, since $\widetilde{K}_{\lambda,\gamma^{0}}(x,y)\subset K(x,y)$ we see from (3.43) that $\tilde{g}_{\lambda,\gamma^{0}}(x,y;u,v)\leq\bar{h}(u,v)+\bar{h}_{\lambda}(u,v)$ where

[TABLE]

and $\bar{h}_{\lambda}(u,v):={\bf 1}((u,v)\in K(x,y)\setminus\widetilde{K}_{\lambda,\gamma^{0}}(x,y))\int_{\widetilde{K}^{c}_{\lambda,\gamma^{0}}(x,y)}\rho(t-u,s-v)\mathrm{d}t\mathrm{d}s.$ Then $\|\bar{h}\|<\infty,\|\bar{h}_{\lambda}\|\to 0$ follow from Lemma 3.1, (3.22)-(3.23). The above facts together with the dominated convergence theorem prove (3.38) for $j=0$ or $\gamma=\gamma^{0}$.

Next, consider (3.38) for $j=1$ or $\gamma>\gamma^{0}$. Let $m_{1}=\tilde{m}_{1}:=\lambda,m_{2}:=\lambda^{\gamma},\tilde{m}_{2}:=\lambda^{\gamma^{0}}$. Using Lemma 3.2, in (3.38) we can replace $\tilde{g}_{\lambda,\gamma}(x,y;\cdot,\cdot)$ of (3.39) by its restriction on $\mathbb{R}\times(0,\lfloor\lambda^{\gamma}y\rfloor/\lambda^{\gamma}]$, or the function

[TABLE]

where

[TABLE]

tends to $a_{\infty}(t-u,s)$ point-wise for any $t\neq u,s\in\mathbb{R},0<v<y$. Hence as $-\lceil\lambda^{\gamma}v\rceil/\lambda^{\gamma^{0}}\to-\infty,\lceil\lfloor\lambda^{\gamma}y\rfloor-\lambda^{\gamma}v\rceil/\lambda^{\gamma^{0}}\to\infty\ (0<v<y)$ we see that $h_{\lambda,\gamma}(x,y;u,v)\to h_{1}(x,y;u,v)$ in (2.4) point-wise for any $(u,v)\in\mathbb{R}^{2},u\not\in\{0,x\},v\not\in\{0,y\}$. Then the $L^{2}(\mathbb{R}^{2})$-convergence in (3.38) for $j=1$ follows similarly as in the case $j=0$ above. The proof of (3.38) for $j=2$ or $\gamma<\gamma^{0}$ is similar using $m_{1}=\lambda,\tilde{m}_{1}=\lambda^{\gamma/\gamma^{0}},m_{2}=\tilde{m}_{2}=\lambda^{\gamma}$ and Lemma 3.2. Theorem 2.2 is proved. $\Box$

3.4 Proof of Theorem 2.3

(i) The convergence in (2.19) for $\gamma>\gamma^{0}_{\rm edge,2}$ follows from the the proof of Theorem 2.2 or the convergence in (3.38) with $\tilde{g}_{\lambda,\gamma}(x,y;\cdot,\cdot)$ replaced by $h_{\lambda,\gamma}(x,y;u,v)$ of (3.44).

Let us prove (2.19) for $\gamma<\gamma^{0}_{\rm edge,2}$. Using Lemma 3.2 we can there replace $S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)$ by

[TABLE]

Split $S^{0}_{\lambda,\gamma}(x,y)=R^{0}_{\lambda,\gamma}(x,y)+R^{+}_{\lambda,\gamma}(x,y)+R^{-}_{\lambda,\gamma}(x,y)$ where

[TABLE]

where $R^{0}_{\lambda,\gamma}(x,y)$ is further rearranged as

[TABLE]

where

[TABLE]

Let us prove that $M^{\pm}_{\lambda,\gamma}(x,y)$ are negligible, viz.,

[TABLE]

(3.45) follows by integral approximation as in the proof of Lemma 3.2. Indeed, using (3.10) the l.h.s. of (3.45) can be evaluated as

[TABLE]

We finally arrive at the main term:

[TABLE]

Let us show the convergence

[TABLE]

Note $|s-v|\geq 1$ in (3.46) and $\min\{|s-v|;s,v\in\mathbb{Z}\}$ is achieved at the lower and upper rows $\{1\leq t\leq\lfloor\lambda x\rfloor,s=1\}$ and $\{1\leq t\leq\lfloor\lambda x\rfloor,s=\lfloor\lambda^{\gamma}y\rfloor\}$ of the rectangle $K_{[\lambda x,\lambda^{\gamma}y]}$. For a large but fixed $L\geq 1$ decompose the l.h.s. of (3.47) as $M_{\lambda,\gamma}(x,y)=\sum_{i=0}^{4}V^{(i)}_{L,\lambda,\gamma}(x,y)$, where

[TABLE]

$V^{(0)}_{L,\lambda,\gamma}(x,y):=M_{\lambda,\gamma}(x,y)-\sum_{i=1}^{4}V^{(i)}_{L,\lambda,\gamma}(x,y)$. Note $V^{(i)}_{L,\lambda,\gamma}(x,y),i=1,2,3,4$ are independent and have similar structure, each consisting of a finite number $L$ of (independent) weighted sums of noise variables $\varepsilon(u,v),1\leq t\leq\lfloor\lambda x\rfloor$ in a neighborhood of the lower ($v=0$) or upper ($v=\lceil\lambda^{\gamma}y\rceil$) boundaries of the rectangle $K_{[\lambda x,\lambda^{\gamma}y]}$; moreover, the distribution of these sums does not depend on $y$. We claim that

[TABLE]

The asymptotic variances in (3.49) are given by

[TABLE]

We omit the proof of (3.49)-(3.51) which reduces to a standard application of Lindeberg’s theorem and the dominated convergence theorem and relies on the fact that the series for $\sigma^{2}_{\rm edge,1}$ in (2.20) absolutely converges:

[TABLE]

The convergence in (3.53) follows from (3.15) and the assumption $Q_{\rm edge,2}<1$. By Slutsky’s theorem, (3.49)-(3.51) imply (3.47), thereby completing the proof of Theorem 2.3 (i). The proof of Theorem 2.3 (ii) is analogous and is omitted. $\Box$

3.5 Proof of Theorem 2.4

For $\gamma<\gamma^{\mathfrak{X}}_{0}=1$ the convergence in (3.47) and the approximation $S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)=M_{\lambda,\gamma}(x,y)+o_{p}(\lambda^{1/2})$ follow as in the proof of Theorem 2.3 (i); particularly, (3.45) holds for $\gamma<1$ in view of (3.13) and $Q_{\rm edge,1}<1$. For $\gamma>1$ the proof of the theorem is analogous using the approximation $S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)=N_{\lambda,\gamma}(x,y)+o_{p}(\lambda^{\gamma/2})$, where

[TABLE]

is the analog of $M_{\lambda,\gamma}(x,y)$ in (3.46) and satisfies

[TABLE]

where $B_{2}$ is a standard Brownian motion. Finally, for $\gamma=1$ we have the approximation $S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)=M_{\lambda,\gamma}(x,y)+N_{\lambda,\gamma}(x,y)+o_{p}(\lambda^{1/2})$ and the joint convergence in (3.47) and (3.54) towards independent BM $B=B_{1}$ and $B_{2}$, thus proving Theorem 2.4. $\Box$

Remark 3.1

It follows from the proofs of Theorems 2.3–2.4 that the FBS limits $B_{1/2,0}$ and $B_{0,1/2}$ in Table 1 (associated with the ‘edge effects’ in Remark 2.2) essentially reduce to the approximation of $S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)$ by sums of horizontal and/or vertical ‘edge processes’ defined below. Particularly,

[TABLE]

where, for each $s\in\mathbb{Z}$ the ‘horizontal process’ $U_{s}=\{U_{s}(t);t\in\mathbb{Z}\}$ ‘located’ on the line $\{(t,s)\in\mathbb{Z}^{2}:t\in\mathbb{Z}\}\subset\mathbb{Z}^{2}$ is defined by

[TABLE]

The process $U_{s}$ is stationary and SRD which follows from

[TABLE]

see (3.53). Moreover, $\sum_{t\in\mathbb{Z}}{\rm Cov}(U_{s}(0),U_{s}(t))=(1/2)\sigma^{2}_{\rm edge,1}$ and the ‘edge processes’ $U_{0}$ and $U_{\lfloor\lambda^{\gamma}y\rfloor}$ in (3.55) ‘located’ on the lower and upper edges of the rectangle $K_{[\lambda x,\lambda^{\gamma}y]}$ become uncorrelated as $\lambda\to\infty$. The above facts may explain the asymptotic variance $\sigma^{2}_{\rm edge,1}$ in (2.20) by approximation in (3.55). The convergence \lambda^{-1/2}\sum_{1\leq t\leq\lfloor\lambda x\rfloor}U_{s}(t)\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\longrightarrow}}\\ \end{array}(1/\sqrt{2})\sigma_{\rm edge,1}B(x) follows similarly as in (3.49). Analogous approximation of $S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)$ by vertical ‘edge processes’ holds in Theorems 2.3-2.4 for $\gamma>\gamma^{\mathfrak{X}}_{0}$.

Remark 3.2

As noted above, FBS $B_{1/2,0}$ and $B_{0,1/2}$ may arise under anisotropic scaling of very simple finitely-dependent RF with ND. Consider a linear RF $\mathfrak{X}$ in (1.6) with coefficients

[TABLE]

Then $S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)=\sum_{1\leq t\leq\lfloor\lambda x\rfloor}\varepsilon(t,1)-\sum_{1\leq t\leq\lfloor\lambda x\rfloor}\varepsilon(t,\lfloor\lambda^{\gamma}y\rfloor+1)$ is the difference of two independent sums of i.i.d.r.v.s, each of which tends to a standard BM under normalization $\lambda^{1/2}$. It easily follows that for each $y>0$ fixed, \lambda^{-1/2}S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\longrightarrow}}\\ \end{array}B^{(0)}(x)-B^{(1)}(x) where $B^{(0)},B^{(1)}$ are independent BM. In a similar way, for any $(x_{i},y_{i})\in\mathbb{R}^{2}_{+},i=1,\cdots,m$ as in (2.22) we have the joint convergence in distribution of the random vector (2\lambda)^{-1/2}\big{(}S^{\mathfrak{X}}_{\lambda,\gamma}(x_{i},y_{i}),i=1,\cdots,m\big{)} towards the Gaussian vector in (2.22) implying the convergence (2\lambda)^{-1/2}S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\longrightarrow}}\\ \end{array}B_{1/2,0}(x,y) for any $\gamma>0$. Note that (3.57) satisfy the vertical ND property in (2.25).

4 Fractionally integrated negative dependent RFs

1. Isotropic fractionally integrated RF. Introduce the (discrete) Laplacian $\Delta Y(t,s):=(1/4)\sum_{|u|+|v|=1}$ $(Y(t+u,s+v)-Y(t,s))$ and a lattice isotropic linear RF

[TABLE]

where $\{\varepsilon(t,s),(t,s)\in\mathbb{Z}^{2}\}$ are standard i.i.d. r.v.s,

[TABLE]

and $p_{j}(u,v)$ are $j$-step transition probabilities of a symmetric nearest-neighbor random walk $\{W_{j};j=0,1,\cdots\}$ on $\mathbb{Z}^{2}$ with equal 1-step probabilities $\mathrm{P}(W_{1}=(u,v)|W_{0}=(0,0))=1/4,|u|+|v|=1$. Note $\sum_{j=0}^{\infty}|\psi_{j}(d)|<\infty,\sum_{j=0}^{\infty}\psi_{j}(d)=0\ (-1/2<d<0),\ \sum_{j=0}^{\infty}\psi_{j}(d)^{2}<\infty\ (0<d<1/2)$ implying

[TABLE]

see also [15, 29]. As shown in the latter papers, for $0<d<1/2$ the linear RF $\mathfrak{X}$ in (4.1) is the unique stationary solution of the fractional equation

[TABLE]

where the operator on the l.h.s. is defined as

[TABLE]

Moreover, for $0<d<1/2$ the RF $\mathfrak{X}$ in (4.1) is LRD with moving-average coefficients satisfying (1.7) with $q_{1}=q_{2}=2(1-d)\in(1,2)$ and a constant angular function $L_{0}$. By stationary solution of (4.4) we mean a covariance stationary RF $\mathfrak{Y}=\{\mathfrak{Y}(t,s);(t,s)\in\mathbb{Z}^{2}\}$, with finite second moment $\mathrm{E}\mathfrak{Y}(t,s)^{2}<\infty$ such that the series $(-\Delta)^{d}\mathfrak{Y}(t,s)=\sum_{(u,v)\in\mathbb{Z}^{2}}b(u,v)\mathfrak{Y}(t-u,s-v)$ converges in mean square and $(-\Delta)^{d}\mathfrak{Y}(t,s)=\varepsilon(t,s)$ holds, for each $(t,s)\in\mathbb{Z}^{2}$. The following proposition extends the above-mentioned result to negative $d\in(-1/2,0)$.

Proposition 4.1

Let $-1/2<d<0$. Then:

(i) RF $\mathfrak{X}$ in (4.1) is a stationary solution of equation (4.5). Moreover, this solution is unique among the class of all linear RFs $\mathfrak{Y}(t,s)=\mu+\sum_{(u,v)\in\mathbb{Z}^{2}}c(u,v)\varepsilon(t-u,s-v),(t,s)\in\mathbb{Z}^{2}$ with $\sum_{(u,v)\in\mathbb{Z}^{2}}|c(u,v)|<\infty$ and $\mu=\mathrm{E}\mathfrak{Y}(t,s)=0$.

(ii) $\mathfrak{X}$ in (4.1) is a ND RF and

[TABLE]

where $A:=\pi^{-1}\Gamma(1-d)/\Gamma(d)<0.$ Particularly, $a(t,s)$ satisfy (1.7) with $q_{1}=q_{2}=2(1-d)\in(2,3),Q=1/(1-d)\in(2/3,1)$ and a constant angular function $L_{0}(z)=A$.

Proof. We use the spectral representation $\varepsilon(t,s)=\int_{\Pi^{2}}\mathrm{e}^{\mathrm{i}(tx+sy)}Z(\mathrm{d}x,\mathrm{d}y),(t,s)\in\mathbb{Z}^{2},$ where $\Pi:=[-\pi,\pi],Z(\mathrm{d}x,\mathrm{d}y)$ is a (random) complex-valued spectral measure, $\overline{Z(\mathrm{d}x,\mathrm{d}y)}=Z(-\mathrm{d}x,-\mathrm{d}y),\mathrm{E}Z(\mathrm{d}x,\mathrm{d}y)=0,\mathrm{E}|Z(\mathrm{d}x,\mathrm{d}y)|^{2}=(2\pi)^{-2}\mathrm{d}x\mathrm{d}y.$ Then $\mathfrak{X}(t,s)$ in (4.1) can be written as

[TABLE]

where $\widehat{a}(x,y)$ is the Fourier transform:

[TABLE]

$\widehat{p}_{1}(x,y):=(1/4)\sum_{|u|+|v|=1}\mathrm{e}^{-\mathrm{i}(ux+vy)}=(\cos x+\cos y)/2$, see ([15], (5.5)). Applying the operator $(-\Delta)^{d}$ (4.5) to $X$ in (4.7) we obtain (-\Delta)^{d}\mathfrak{X}(t,s)=\int_{\Pi^{2}}\mathrm{e}^{\mathrm{i}(tx+sy)}\big{(}\sum_{(u,v)\in\mathbb{Z}^{2}}\mathrm{e}^{-\mathrm{i}(ux+vy)}b(u,v)\big{)}\widehat{a}(x,y)Z(\mathrm{d}x,\mathrm{d}y)=\int_{\Pi^{2}}\mathrm{e}^{\mathrm{i}(tx+sy)}\widehat{b}(x,y)\widehat{a}(x,y)Z(\mathrm{d}x,\mathrm{d}y)=\varepsilon(t,s) since $\widehat{b}(x,y)=1/\widehat{a}(x,y),$ the series $\sum_{(u,v)\in\mathbb{Z}^{2}}\mathrm{e}^{-\mathrm{i}(ux+vy)}b(u,v)=\widehat{b}(x,y)$ converges in $L^{2}(\Pi^{2})$ and $\widehat{a}(x,y)$ is bounded on $\Pi^{2}$. To show the uniqueness part, let $\mathfrak{Y}(t,s)=$ $\sum_{(u,v)\in\mathbb{Z}^{2}}c(u,v)\varepsilon(t-u,s-v)$ be another solution having spectral representation $\mathfrak{Y}(t,s)=\int_{\Pi^{2}}\mathrm{e}^{\mathrm{i}(tx+sy)}\widehat{c}(x,y)$ $Z(\mathrm{d}x,\mathrm{d}y)$ where $\widehat{c}(x,y)$ is a bounded function on $\Pi^{2}$. Then  $(-\Delta)^{d}(\mathfrak{X}(t,s)-\mathfrak{Y}(t,s))=\int_{\Pi^{2}}\mathrm{e}^{\mathrm{i}(tx+sy)}\widehat{b}(x,y)$ $(\widehat{a}(x,y))-\widehat{c}(x,y))Z(\mathrm{d}x,\mathrm{d}y)=0$ or $\int_{\Pi^{2}}|\widehat{b}(x,y)|^{2}|\widehat{c}(x,y)-\widehat{a}(x,y)|^{2}\mathrm{d}x\,\mathrm{d}y=0$ implying $\widehat{c}(x,y)=\widehat{a}(x,y)$ a.e. on $\Pi^{2}$ since $|\widehat{b}(x,y)|^{2}=|1-\widehat{p}_{1}(x,y)|^{2d}>0$ a.e. on $\Pi^{2}$. Therefore, $c(t,s)=a(t,s)$ and $\mathfrak{Y}(t,s)=\mathfrak{X}(t,s),(t,s)\in\mathbb{Z}^{2}$.

(ii) The ND property of $\mathfrak{X}$ (4.1) follows from the zero-sum condition in (4.3). The proof of (4.6) using the Moivre-Laplace theorem carries over from ([15], proof of Proposition 5.1) to the case $-1/2<d<0$ virtually without any change. $\Box$

Remark 4.1

It follows from (4.7) that RF $\mathfrak{X}$ has an explicit spectral density $f(x,y)=(2\pi)^{-2}|\widehat{a}(x,y)|^{2}=2^{-2d}|(1-\cos x)+(1-\cos y)|^{-2d},(x,y)\in\Pi^{2}$ which vanishes as ${\rm const}\,(x^{2}+y^{2})^{-2d}\to 0$ as $x^{2}+y^{2}\to 0$.

2. Anisotropic ND fractionally integrated RF. Following [29] consider the ‘discrete heat operator’ $\Delta_{1,2}\mathfrak{X}(t,s)$ $:=\mathfrak{X}(t,s)-\theta\mathfrak{X}(t-1,s)-\frac{1-\theta}{2}(\mathfrak{X}(t-1,s+1)+\mathfrak{X}(t-1,s-1)),0<\theta<1$ and a corresponding ‘fractional power’ defined as

[TABLE]

where

[TABLE]

where $q_{\theta}(u,v)$ are $u$-step transition probabilities of a random walk $\{W_{u};u=0,1,\cdots\}$ on $\mathbb{Z}$ with 1-step probabilities $\mathrm{P}(W_{1}=v|W_{0}=0)=\theta$ if $v=0$, $=(1-\theta)/2$ if $v=\pm 1$. As shown in [29], $\sum_{(u,v)\in\mathbb{Z}^{2}}b(u,v)^{2}<\infty$ and

[TABLE]

where

[TABLE]

and $L_{0}(z;d,\theta),z\in[-1,1]$ is a bounded continuous function. Similarly to (4.1), the inverse operator can be defined by $\Delta_{1,2}^{-d}\mathfrak{X}(t,s)\ :=\ \sum_{(u,v)\in\mathbb{Z}_{+}\times\mathbb{Z}}a(u,v)\mathfrak{X}(t-u,s-v),$ where

[TABLE]

are obtained from $b(u,v)$ in (4.10) with $d$ replaced by $-d$. By direct inspection of the proof of Proposition 4.1 in [29] we find that $a(t,s)$ also satisfy (4.11)-(4.12) with $d$ replaced by $-d$, for $-3/4<d<0$, in other words, the linear RF $\mathfrak{X}$ in (4.1) with moving-average coefficients in (4.13) satisfies the assumptions in (1.7) with $q_{1}=(3/2)-d,q_{2}=2q_{1}=2((3/2)-d)$ and

[TABLE]

Moreover, the above $\mathfrak{X}$ can be regarded as a stationary solution (in the sense as explained before Proposition 4.1) of the fractional equation:

[TABLE]

We arrive at the following proposition whose proof is similar to that of Proposition 4.1 and we omit the details.

Proposition 4.2

Let $-3/4<d<0$. Then:

(i) RF $\mathfrak{X}$ in (4.1) with coefficients $a(u,v)$ in (4.13) is a stationary solution of equation (4.14). Moreover, this solution is unique among the class of all linear RFs $\mathfrak{Y}(t,s)=\mu+\sum_{(u,v)\in\mathbb{Z}^{2}}c(u,v)\varepsilon(t-u,s-v),(t,s)\in\mathbb{Z}^{2}$ with $\sum_{(u,v)\in\mathbb{Z}^{2}}|c(u,v)|<\infty$ and $\mu=\mathrm{E}\mathfrak{Y}(t,s)=0$.

(ii) The above $\mathfrak{X}$ is a ND RF and $a(u,v)$ satisfy (1.7) with $q_{1}=(3/2)-d,q_{2}=2q_{1}=2((3/2)-d)$ and a continuous angular function $L_{0}(z)=L_{0}(z;\theta,-d),z\in[-1,1]$ as defined in (4.12).

3. Separately fractionally integrated RF. Several works discussed RFs where the operators of fractional integration are applied separately w.r.t. each of the two coordinates on the plane. The moving-average coefficients of such RFs have a product form which is very different from (1.7) and apparently does not allow for scaling transition. See ([29], Remark 4.1), [5], ([36], Remark 4.1). Gaussian LRD RFs whose spectral density has a product form appear in [21, 31]. Particularly, [4, 13] discussed fractionally integrated RFs satisfying the equation

[TABLE]

where $\nabla_{1}\mathfrak{X}(t,s):=\mathfrak{X}(t,s)-\mathfrak{X}(t-1,s),\nabla_{2}\mathfrak{X}(t,s):=\mathfrak{X}(t,s)-\mathfrak{X}(t,s-1)$ are difference operators and $0<d_{1},d_{2}<1/2$ are parameters. Stationary solution of (4.15) is a moving-average RF in $\mathfrak{X}(t,s)=\sum_{(u,v)\in\mathbb{Z}_{+}^{2}}a(u,v)\varepsilon(t-u,v-s)$ with coefficients $a(u,v):=\psi_{u}(-d_{1})\psi_{v}(-d_{2}).$ Equation (4.15) and its solution can be extended to any $d_{i}\in(-1/2,1/2),i=1,2$. Since the covariance function of this RF factorizes into the product of two covariances of ARFIMA processes with respective parameters $d_{1},d_{2}$, it follows that the RF $\mathfrak{X}$ in (4.15) is LRD if $d_{1}\vee d_{2}>0$ and ND if $d_{1}\vee d_{2}\leq 0,d_{1}\wedge d_{2}<0$. Moreover, following ([29], Theorem 3.1) and Theorem 2.2 of the present paper, one can show that for any $\gamma>0$ and any $|d_{i}|<1/2,i=1,2$ the (normalized) partial sums process of RF $\mathfrak{X}$ in (4.15) tends to a FBS depending on $d_{1},d_{2}$ only, viz., \lambda^{-H_{1}-\gamma H_{2}}S^{\mathfrak{X}}_{\lambda,\gamma}(x,y)\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\longrightarrow}}\\ \end{array}c(d_{1})c(d_{2})B_{H_{1},H_{2}}(x,y), where $H_{i}=d_{i}+1/2$ and $c(d_{i})>0$ are some constants. As a consequence, the separately fractionally integrated RF in (4.15) does not exhibit scaling transition in both (LRD and ND) parameter regions.

[8] discussed separately fractionally integrated Gegenbauer RF satisfying

[TABLE]

where $\nabla_{u_{i}}^{d_{i}}:=(1-2u_{i}(1-\nabla_{i})+(1-\nabla_{i})^{2})^{d_{i}},|u_{i}|<1,d_{i}\in(0,1/2),i=1,2$ are parameters. [8] proved that $\mathfrak{X}$ in (4.16) is a LRD RF which exhibits seasonal or cyclical dependence, due to the fact that its spectral density is positive and continuous at the origin and is singular on two lines parallel to the coordinate axes. We conjecture that all scaling limits of $\mathfrak{X}$ in (4.16) agree with the Brownian Sheet $B_{1/2,1/2}$ and the above RF does not exhibit scaling transition. See [24] for related results in the time series setting.

Remark 4.2

[20] discuss a wide class of fractionally integrated Gaussian RFs on $\mathbb{R}^{\nu},\nu\geq 1$, defined via Fourier transform, together with numerous references. Scaling properties of these RFs seem to be less investigated.

5 Concluding remarks

The present paper and [29] provide a nearly complete description, summarised in Table 1, of anisotropic scaling limits on inflated rectangles of linear LRD, SRD and ND RFs on $\mathbb{Z}^{2}$ 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 rectangle. We expect that these results can be generalized to linear RFs in higher dimensions $\nu\geq 3$, as well as to linear RFs with infinite variance, although the description of the anisotropic scaling limits in [36] for LRD RFs and $\nu=3$ is more complicated. We also expect that our results remain valid, in some sense, for more general sampling regions similar to [16] and ‘anisotropically inflated’ by scaling factors $\lambda$ and $\lambda^{\gamma}$ in the horizontal and vertical directions.

The scaling results in Table 1 strongly rely on scaling properties of moving-average coefficients $a(t,s)$ in (1.7). See ([36], Remark 4.1). In the Gaussian case ($\varepsilon(0,0)\sim N(0,1)$) the RF $\mathfrak{X}$ (1.6) is completely described by its spectral density (which exists for $Q<2$ due to square summability of $a(t,s)$, see [29]). For Gaussian LRD RFs on $\mathbb{Z}^{2}$ similar scaling results were obtained in [31], assuming that the spectral density has an asymptotic form at the origin which resembles that of the moving-average coefficients at infinity. A rigorous statement relating (1.7) to the behavior of its Fourier transform at the origin (Abelian/Tauberian theorem [22]) is an interesting open problem; see ([36], Remark 2.2).

A direction on the plane (a line passing through the origin) along which the moving-average coefficients $a(t,s)$ decay at the smallest rate may be called the dependence axis of linear RF $\mathfrak{X}$ in (1.6). Consider the coefficients $a(t,s)$ as in (1.7) where $L_{0}\equiv 1$, say. Then $a(t,0)\sim|t|^{-q_{1}},a(0,s)\sim|s|^{-q_{2}}$ and the dependence axis of the corresponding RF $\mathfrak{X}$ agrees with the horizontal axis if $q_{1}<q_{2}$ and with the vertical axis if $q_{1}>q_{2}$. Since the scaling in (2.19) is parallel to the coordinate axes, we may say that for RF $\mathfrak{X}$ in (1.6)-(1.7) the scaling is congruous with the dependence axis of $\mathfrak{X}$ and the results in Table 1 refer to this situation. This raises the question what happens when the dependence axis does not agree with the coordinate axes (the case of incongruous scaling), particularly, the question about the scaling transition and the scaling transition point $\gamma^{\mathfrak{X}}_{0}$ under incongruous scaling. A linear RF with ‘oblique’ dependence axis can be obtained by a ‘rotation’ of $a(t,s)$ in (1.7) or a more general linear transformation of the plane. We conjecture that the fact of ‘oblique’ dependence axis or incongruous scaling dramatically changes the scaling transition so that the scaling transition point is always 1: $\gamma^{\mathfrak{X}}_{0}=1$ independently of $q_{1},q_{2}$, and the unbalanced limits are generally different from the corresponding limits in the congruous scaling case (in the LRD region $1<Q<2$ this conjecture was recently confirmed in [30]).

Finally, we expect that many results in Table 1 can be carried over to RFs indexed by continuous argument $(t,s)\in\mathbb{R}^{2}$. Particularly, this remark applies to stationary Gaussian RFs which are completely described by spectral measure on $\mathbb{R}^{2}$. A popular class of nongaussian RFs with continuous argument is shot-noise, see [1], [11] and the references therein. However, the scaling diagram of LRD shot-noise RF may be very different from Table 1. See [28]. These and further open problems are discussed in more detail in the recent review paper [37].

Acknowledgments

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