# Scaling transition for singular linear random fields on $\mathbb{Z}^{2}$:
spectral approach

Donatas Surgailis

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

###### Abstract

We study partial sums limits of linear random fields $X$ on $\mathbb{Z}^{2}$
with spectral density $f(\boldsymbol{x})$ tending to $\infty,\,0$ or to both
(along different subsequences) as $\boldsymbol{x}\to(0,0)$. The above
behaviors are termed (spectrum) long-range dependence, negative dependence,
and long-range negative dependence, respectively, and assume an anisotropic
power-law form of $f(\boldsymbol{x})$ near the origin. The partial sums are
taken over rectangles whose sides increase as $\lambda$ and
$\lambda^{\gamma}$, for any fixed $\gamma>0$. We prove that for above $X$ the
partial sums or scaling limits exist for any $\gamma>0$ and exhibit a scaling
transition at some $\gamma=\gamma_{0}>0$; moreover, the ‘unbalanced’ scaling
limits ($\gamma\neq\gamma_{0}$) are Fractional Brownian Sheet with Hurst
parameters taking values from $[0,1]$. The paper extends [24, 19, 28] to the
above spectrum dependence conditions and/or more general values of Hurst
parameters.

Keywords: Linear random field, Gaussian random field, spectral density, long-
range dependence, negative dependence, long-range negative dependence,
hyperbolic dependence, self-similarity, anisotropic scaling, scaling
transition, fractional Brownian sheet,

##  1 Introduction

A stationary random field (RF)
$X=\\{X(\boldsymbol{t});\boldsymbol{t}\in\mathbb{Z}^{d}\\}$ with finite second
moment and covariance $r(\boldsymbol{t})={\rm Cov}(X({\bf
0}),X(\boldsymbol{t}))$ is said (i) long-range dependent (LRD) if
$\sum_{\boldsymbol{t}\in\mathbb{Z}^{d}}|r(\boldsymbol{t})|=\infty$; (ii)
short-range dependent (SRD) if
$\sum_{\boldsymbol{t}\in\mathbb{Z}^{d}}|r(\boldsymbol{t})|<\infty,\ $
$\sum_{\boldsymbol{t}\in\mathbb{Z}^{d}}r(\boldsymbol{t})\neq 0$, and (iii)
negatively dependent (ND) if
$\sum_{\boldsymbol{t}\in\mathbb{Z}^{d}}|r(\boldsymbol{t})|<\infty,\
\sum_{\boldsymbol{t}\in\mathbb{Z}^{d}}r(\boldsymbol{t})=0$ (for RF indexed by
continuous argument, the above concepts are analogously defined with sums
replaced by integrals over $\mathbb{R}^{d}$). The above classification, albeit
not completely satisfactory, is important in limit theorems, see [13, 16, 28].
Related but not equivalent definitions of LRD, SRD and ND are given in terms
of spectrum or moving-average coefficients of RF. In the sequel we refer to
the above covariance or moving-average characterizations as spatial domain
properties, to be distinguished from frequency domain (spectrum)
characterizations used in the literature. A very general form of such
frequency domain concepts assumes the existence of spectral density
$f(\boldsymbol{u}),\boldsymbol{u}\in\Pi^{d}:=[-\pi,\pi]^{d}$ which is bounded
outside of the origin and such that (i’) $\lim_{\boldsymbol{u}\to{\bf
0}}f(\boldsymbol{u})=\infty$ (spectrum LRD), (ii’)
$\lim_{\boldsymbol{u}\to{\bf 0}}f(\boldsymbol{u})>0$ (spectrum SRD), and
(iii’) $\lim_{\boldsymbol{u}\to{\bf 0}}f(\boldsymbol{u})=0$ (spectrum ND).
Moreover, in cases (i’) and (iii’) it is often additionally assumed that
spectral density varies regularly as $\boldsymbol{u}\to 0$ [13, 22, 25]. It is
clear that in dimensions $d\geq 2$ a function can increase regularly to
$\infty$ along one direction and vanish along another one. Such a behavior is
not reflected in (i’)-(iii’), calling for a new concept (iv’)
$0=\liminf_{\boldsymbol{u}\to{\bf
0}}f(\boldsymbol{u})<\limsup_{\boldsymbol{u}\to{\bf
0}}f(\boldsymbol{u})=\infty$ which we term spectrum long-range negative
dependence  (spectrum LRND). Properties (i’), (iii’) and (iv’) can be jointly
dubbed as singular  behaviors, the singularity limited to the origin and
including both infinite and/or zero limits.

The present paper studies anisotropic partial sums limits for a class of
stationary linear RFs $X$ on $\mathbb{Z}^{2}$ with spectrum LRD/ND/LRND
properties. The partial sums

$S_{\lambda,\gamma}(\boldsymbol{x}):=\ \sum_{\boldsymbol{t}\in K_{[\lambda
x_{1},\lambda^{\gamma}x_{2}]}}X(\boldsymbol{t}),\qquad\boldsymbol{x}=(x_{1},x_{2})\in\mathbb{R}^{2}_{+}$
(1.1)

are taken over rectangles $K_{[\lambda
x_{1},\lambda^{\gamma}x_{2}]}:=\\{\boldsymbol{t}\in\mathbb{Z}^{2}:1\leq
t_{1}\leq\lambda x_{1},1\leq
t_{2}\leq\lambda^{\gamma}x_{2}\\}\subset\mathbb{Z}^{2}$ whose sides increase
as $O(\lambda)$ and $O(\lambda^{\gamma})$ as $\lambda\to\infty$, for a fixed
$\gamma>0$. Our main object are anisotropic scaling limits

$d^{-1}_{\lambda,\gamma}S_{\lambda,\gamma}(\boldsymbol{x})\
\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\longrightarrow}}\\\
\end{array}\
V_{\gamma}(\boldsymbol{x}),\quad\boldsymbol{x}\in\mathbb{R}^{2}_{+},\quad\lambda\to\infty,$
(1.2)

where $d_{\lambda,\gamma}\to\infty$ is a normalization. It was proved in [24,
23, 18, 19, 4, 20, 28] and other works that for a large class of RFs
nontrivial scaling limits $V_{\gamma}(\boldsymbol{x})$ in (1.2) exist for any
$\gamma>0$. Moreover, the limit family $\\{V_{\gamma};\gamma>0\\}$ may exhibit
a trichotomy called scaling transition at some (nonrandom) point
$\gamma=\gamma_{0}>0$ meaning that $V_{\gamma}=V_{\pm}\
(\gamma\lessgtr\gamma_{0})$ do not depend on $\gamma$ and are different; see
Definition 1. For linear RFs, these works can be summarized as saying that a
large class of RFs on $\mathbb{Z}^{2}$ parade two types of the limit behavior
in (1.2): either (I) $V_{\gamma}$ is a standard Brownian Sheet $B_{1/2,1/2}$
for any $\gamma>0$, or (II) scaling transition at some $\gamma_{0}>0$ exists
such that all unbalanced limits $V_{\gamma},\gamma\neq\gamma_{0}$ are
Fractional Brownian Sheet (FBS) $B_{H_{1},H_{2}}$ with Hurst parameters
$(H_{1},H_{2})=(H^{+}_{1},H^{+}_{2})\,(\gamma>\gamma_{0}),\
=(H^{-}_{1},H^{-}_{2})\,(\gamma<\gamma_{0})$,
$(H^{+}_{1},H^{+}_{2})\neq(H^{-}_{1},H^{-}_{2})$. Behavior (I) is rather
standard under SRD conditions, see [6, 7, 23] and the references therein.
Behavior (II) is more interesting and was established under LRD and ND
conditions in [24, 28]. However, these results exclude LRND singularity and
some particular (boundary) values of parameters.

|
---|---
|

Figure 1. Level graphs of spectral density under LRD, ND, LRND and ‘hyperbolic
dependence’ conditions. a) $f(\boldsymbol{x})=\rho^{-1}_{2}(\boldsymbol{x})$,
$\upsilon_{1}=0.8,\upsilon_{2}=1.2$; b)
$f(\boldsymbol{x})=\rho(\boldsymbol{x}),\upsilon_{1}=0.5,\upsilon_{2}=1$; c)
$f(\boldsymbol{x})=\rho^{-1}(\boldsymbol{x})|x_{2}/\rho(\boldsymbol{x})^{1/\upsilon_{2}}|^{0.5},\upsilon_{1}=0.5,\upsilon_{2}=1$;
d) $f(\boldsymbol{x})=|x_{1}|^{-0.2}|x_{2}|^{-.5}$

The present paper extends [24, 19, 28] and some other work on type (II)
behavior for linear RFs $X$ under spectrum LRD, ND, and LRND conditions. It
appears that the spectral approach used in this work is more efficient and
simple compared to ‘time-domain’ approach in [19, 28] and other papers;
moreover, it can be used to obtain the covariance structure of the scaling
limits in a non-Gaussian context as well. It is assumed that $X$ have a
moving-average (MA) representation

$X(\boldsymbol{t})=\sum_{\boldsymbol{s}\in\mathbb{Z}^{2}}a(\boldsymbol{t}-\boldsymbol{s})\varepsilon(\boldsymbol{s}),\qquad\boldsymbol{t}\in\mathbb{Z}^{2},$
(1.3)

with standardized i.i.d. innovations
$\\{\varepsilon(\boldsymbol{s});\boldsymbol{s}\in\mathbb{Z}^{2}\\},\mathrm{E}\varepsilon(\boldsymbol{s})=0,\mathrm{E}\varepsilon(\boldsymbol{s})^{2}=1$
and deterministic coefficients
$a(\boldsymbol{t}),\boldsymbol{t}\in\mathbb{Z}^{2}.$ The latter coefficients
satisfy the necessary condition
$\sum_{\boldsymbol{t}\in\mathbb{Z}^{2}}a(\boldsymbol{t})^{2}<\infty$ for the
convergence of (1.3) but all other conditions for various behaviors in (1.2)
in this paper refer to the spectral density of $X$ written as the squared
Fourier transform
$f(\boldsymbol{x})=(2\pi)^{2}|\widehat{a}(\boldsymbol{x})|^{2},$
$\widehat{a}(\boldsymbol{x}):=(2\pi)^{-2}\sum_{\boldsymbol{t}\in\mathbb{Z}^{2}}\mathrm{e}^{\mathrm{i}\boldsymbol{t}\cdot\boldsymbol{x}}a(\boldsymbol{t}),\
\boldsymbol{x}\in\Pi^{2}$ of MA coeffiecients. For identification of the
covariance structure of the limit Gaussian RF in (1.2), the basic assumption
is that $f(\boldsymbol{x})$ behaves like a power function

$\displaystyle\rho(\boldsymbol{x}):=|x_{1}|^{\upsilon_{1}}+|x_{2}|^{\upsilon_{2}},\qquad\boldsymbol{x}=(x_{1},x_{2})\in\mathbb{R}^{2}$
(1.4)

at the origin, where $\upsilon_{i}>0,i=1,2$ are given parameters, or, more
precisely,

$f(\boldsymbol{x})\sim
L(\boldsymbol{x})/\rho(\boldsymbol{x})\quad\text{(`spectrum
LRD')}\quad\text{and}\quad f(\boldsymbol{x})\sim
L(\boldsymbol{x})\rho(\boldsymbol{x})\quad\text{(`spectrum ND')},$ (1.5)

where $L(\boldsymbol{x})$ is an ‘angular function’ satisfying some boundedness
conditions. Clearly, the first relation in (1.5) implies spectrum LRD provided
$L(\boldsymbol{x})$ is bounded from below, while the second relation relation
in (1.5) implies spectrum ND provided $L(\boldsymbol{x})$ is bounded from
above. The above choice of $\rho(\boldsymbol{x})$ as the ‘radial function’ is
rather flexible: we can use other forms of (1.4), e.g. with

$\displaystyle\rho_{p}(\boldsymbol{x}):=(|x_{1}|^{p\upsilon_{1}}+|x_{2}|^{p\upsilon_{2}})^{1/p},\quad
p>0,\quad\boldsymbol{x}=(x_{1},x_{2})\in\mathbb{R}^{2}$ (1.6)

instead of (1.4) as well, since the change of $\rho(\boldsymbol{x})$ to
$\rho_{p}(\boldsymbol{x})$ amounts to a change of the angular function which
does not affect the limit results. Spectrum LRND may arise if these
boundedness conditions on $L(\boldsymbol{x})$ are violated, e.g. when
$L(\boldsymbol{x})$ decreases as $|x_{i}|^{\mu_{i}}$ with exponent $\mu_{i}>0$
when $x_{i}\to 0$ in the first relation in (1.5). We show that in this case
the limits in (1.2) may exist, together with a scaling transition and the
limit Gaussian RF depends on parameters $\upsilon_{i},\mu_{i},i=1,2$. We also
consider the case when the spectral density asymptotically factorizes into a
product of power functions $|x_{i}|^{-\upsilon_{i}},i=1,2$. This form of
singularity (studied in [24, 6] and termed hyperbolic dependence in the
present paper) allows for spectrum LRD, ND, or LRND depending on the sign of
$\upsilon_{i}$, but leads to very different limit results (the absence of
scaling transition). Fig. 1 illustrates different types of singular spectral
densities studied in this paper. We also expect that our results can be
extended to RFs on $d$-dimensional lattice, $d\geq 3$ arbitrary although the
description of the limit RFs is more complicated, see [4, 27].

The rest of the paper is organized as follows. Sec. 2 provides rigorous
assumptions on $f(\boldsymbol{x})$ and some preliminary facts used in the
subsequent text. Sections 3-5 contain the main Theorems 1, 2, 3 and 4, under
respective LRD, ND, LRND and hyperbolic dependence premises.

Notation. In this paper, $\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
fdd}}{{\longrightarrow}}\\\
\end{array},\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{=}}\\\
\end{array},$ and $\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
fdd}}{{\neq}}\\\ \end{array}$, denote respectively the weak convergence,
equality, and inequality, of finite dimensional distributions. $C$ stands for
a generic positive constant which may assume different values at various
locations and whose precise value has no importance.
$\mathbb{R}_{+}:=(0,\infty),\,\mathbb{R}^{2}_{0}:=\mathbb{R}^{2}\setminus\\{(0,0)\\},\Pi:=[-\pi,\pi],\,|\boldsymbol{x}|:=|x_{1}|+|x_{2}|,\boldsymbol{x}\cdot\boldsymbol{y}:=x_{1}y_{1}+x_{2}y_{2},\boldsymbol{x}=(x_{1},x_{2}),\boldsymbol{y}=(y_{1},y_{2}),{\bf
1}:=(1,1),{\bf 0}:=(0,0)$. $\mathbb{I}(A)$ denotes the indicator function of a
set $A$.

##  2 Preliminaries and assumptions

######  Definition 1.

[24, 23, 19] We say that a stationary RF
$X=\\{X(\boldsymbol{t});\boldsymbol{t}\in\mathbb{Z}^{2}\\}$ exhibits scaling
transition if non-trivial limits in (1.2) exist for any $\gamma>0$ and there
exists a $\gamma_{0}>0$ such that

$V_{\gamma}\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{=}}\\\
\end{array}V_{+}\ \ (\gamma>\gamma_{0}),\qquad
V_{\gamma}\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{=}}\\\
\end{array}V_{-}\ \ (0<\gamma<\gamma_{0}),\qquad
V_{+}\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\neq}}\\\
\end{array}aV_{-}\ \ (\forall\,a>0).$ (2.1)

In such a case, RFs $V_{\pm}$ will be called the unbalanced limits, and RF
$V_{0}$ the well-balanced limit of $X$.

A weaker version of Definition 1 in [7] requires that $V_{\gamma}$ is
independent of $\gamma$ for $\gamma>\gamma_{0}$ and $\gamma<\gamma_{0}$ in a
small neighborhood of $\gamma_{0}$. There exist RFs which exhibit scaling
transition in the above weaker sense at several (possibly, infinite) number of
points $\gamma>0$ [18, 7]. However, for RFs discussed in this paper Definition
1 suffices.

Recall that the covariance function of stationary zero-mean RF
$X=\\{X(\boldsymbol{t});\boldsymbol{t}\in\mathbb{Z}^{2}\\}$ is the Fourier
transform of its spectral density, viz., $r(\boldsymbol{t}):=\mathrm{E}X({\bf
0})X(\boldsymbol{t})=\int_{\Pi^{2}}\mathrm{e}^{\mathrm{i}\boldsymbol{t}\cdot\boldsymbol{u}}f(\boldsymbol{u})\mathrm{d}\boldsymbol{u},\boldsymbol{t}\in\mathbb{Z}^{2}$.
Whence, the covariance function of the normalized partial sums RF in (1.1)
writes as

$\displaystyle R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle:=$ $\displaystyle
d^{-2}_{\lambda,\gamma}\mathrm{E}S_{\lambda,\gamma}(\boldsymbol{x})S_{\lambda,\gamma}(\boldsymbol{y})\
=\
d^{-2}_{\lambda,\gamma}\int_{\Pi^{2}}\prod_{i=1}^{2}D_{[\lambda^{\gamma_{i}}x_{i}]}(u_{i})\overline{D_{[\lambda^{\gamma_{i}}y_{i}]}(u_{i})}f(\boldsymbol{u})\mathrm{d}\boldsymbol{u},\quad\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^{2}_{+},$
(2.2)

where $(\gamma_{1},\gamma_{2}):=(1,\gamma)$ and
$D_{n}(u):=\sum_{t=1}^{n}\mathrm{e}^{\mathrm{i}tu}=(\mathrm{e}^{\mathrm{i}u}-\mathrm{e}^{\mathrm{i}(n+1)u})/(1-\mathrm{e}^{\mathrm{i}u}),u\in\Pi$
(the real-valued function $D_{n}(u)\mathrm{e}^{-\mathrm{i}(n+1)u/2}$ is the
Dirichlet kernel).

######  Proposition 1.

Let $X$ be a linear RF in (1.3) with spectral density $f$ and
$S_{\lambda,\gamma}$ be the partial sum RF in (1.1), $\gamma>0$. Assume that

$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})\ \to\
R_{\gamma}(\boldsymbol{x},\boldsymbol{y}),\quad\lambda\to\infty\quad(\forall\,\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^{2}_{+})$
(2.3)

and

$|g_{\lambda}|_{\infty}:=\sup_{\boldsymbol{u}\in\mathbb{Z}^{2}}\Big{|}\sum_{\boldsymbol{t}\in
K_{[\lambda
x_{1},\lambda^{\gamma}x_{2}]}}a(\boldsymbol{t}-\boldsymbol{u})\Big{|}=o(d_{\lambda,\gamma}),\qquad\lambda\to\infty.$
(2.4)

Then (1.2) holds, where $V_{\gamma}$ is a Gaussian RF on $\mathbb{R}^{2}_{+}$
with zero mean $\mathrm{E}V_{\gamma}(\boldsymbol{x})=0$ and covariance
$\mathrm{E}V_{\gamma}(\boldsymbol{x})V_{\gamma}(\boldsymbol{y})=R_{\gamma}(\boldsymbol{x},\boldsymbol{y}),\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^{2}_{+}$.

Proof. The finite-dimensional convergence in (1.2) is equivalent to one-
dimensional convergence ${\cal
S}_{\lambda,\gamma}:=d^{-1}_{\lambda,\gamma}\sum_{j=1}^{m}\theta_{j}$
$S_{\lambda,\gamma}(\boldsymbol{x}_{j})\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\
\end{array}\sum_{j=1}^{m}\theta_{j}V_{\gamma}(\boldsymbol{x}_{j})=:{\cal
V}_{\gamma}$ for any
$\theta_{j}\in\mathbb{R},\boldsymbol{x}_{j}\in\mathbb{R}^{2}_{+},j=1,\cdots,m,m\geq
1$. From (2.3) we have $\mathrm{E}({\cal
S}_{\lambda,\gamma})^{2}\to\sum_{j,j^{\prime}=1}^{m}\theta_{j}\theta_{j^{\prime}}R_{\gamma}(\boldsymbol{x}_{j},\boldsymbol{x}_{j^{\prime}})=:{\cal
R}_{\gamma}\geq 0$. Let ${\cal R}_{\gamma}>0$ then ${\cal
S}_{\lambda,\gamma}=d^{-1}_{\lambda,\gamma}\sum_{\boldsymbol{u}\in\mathbb{Z}^{2}}g_{\lambda}(\boldsymbol{u})\varepsilon(\boldsymbol{u})$
is a normalized weighted linear form in i.i.d. r.v.s with weights
$g_{\lambda}(\boldsymbol{u}):=\sum_{j=1}^{m}\theta_{j}\sum_{\boldsymbol{t}\in
K_{[\lambda
x_{1j},\lambda^{\gamma}x_{2j}]}}a(\boldsymbol{t}-\boldsymbol{u}),\boldsymbol{x}_{j}=(x_{1j},x_{2j})$.
Relation (2.4) implies the Lindeberg condition, viz., for any $\tau>0$

$\sum_{\boldsymbol{u}\in\mathbb{Z}^{2}}g^{2}_{\lambda}(\boldsymbol{t})\mathrm{E}[\varepsilon(\boldsymbol{u})^{2}\mathbb{I}(|g_{\lambda}(\boldsymbol{u})\varepsilon(\boldsymbol{u})|>\tau
d_{\lambda,\gamma})]=o(d^{2}_{\lambda,\gamma}),\qquad\lambda\to\infty$ (2.5)

since
$\mathrm{E}\varepsilon(\boldsymbol{u})^{2}\mathbb{I}(|g_{\lambda}(\boldsymbol{u})\varepsilon(\boldsymbol{u})|>\tau
d_{\lambda,\gamma})\leq\mathrm{E}\varepsilon(\boldsymbol{u})^{2}\mathbb{I}\big{(}|\varepsilon(\boldsymbol{u})|>\tau
d_{\lambda,\gamma}/|g_{\lambda}|_{\infty}\big{)}\to 0$ in view of (2.4) and
$\mathrm{E}\varepsilon(\boldsymbol{0})^{2}<\infty$. Thus, ${\cal
S}_{\lambda,\gamma}\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\ \end{array}N(0,{\cal R}_{\gamma})$, in other words,
the distribution of
$(d^{-1}_{\lambda,\gamma}S_{\lambda,\gamma}(\boldsymbol{x}_{j});j=1,\cdots,m)$
tends to a Gaussian distribution on $\mathbb{R}^{m}$ with mean zero and
covariance matrix
$(R_{\gamma}(\boldsymbol{x}_{j},\boldsymbol{x}_{j^{\prime}}))_{j,j=1,\cdots,m}$,
proving the proposition. $\Box$

Let
$\upsilon_{i}>0,i=1,2,\Upsilon:=\frac{1}{\upsilon_{1}}+\frac{1}{\upsilon_{2}}$
and $\rho(\boldsymbol{x}),\rho_{p}(\boldsymbol{x})$ be as in (1.4), (1.6). We
note the elementary inequality: for any $p>0$ there exist constants
$0<C_{1}\leq C_{2}<\infty$ such that

$C_{-}\rho(\boldsymbol{x})\leq\rho_{p}(\boldsymbol{x})\leq
C_{+}\rho(\boldsymbol{x}),\qquad(\forall\,\boldsymbol{x}\in\mathbb{R}^{2}).$
(2.6)

We also use the fact [19] that for any $\delta>0,w>0$

$\int_{|\boldsymbol{x}|<\delta}\rho(\boldsymbol{x})^{-w}\mathrm{d}\boldsymbol{x}<\infty\
\Longleftrightarrow\
w<\Upsilon,\qquad\int_{|\boldsymbol{x}|>\delta}\rho(\boldsymbol{x})^{-w}\mathrm{d}\boldsymbol{x}<\infty\
\Longleftrightarrow\ w>\Upsilon.$ (2.7)

Following [20], we say that a measurable function
$L:\mathbb{R}^{2}_{0}\to\mathbb{R}$ is _generalized invariant_ if the function
$\boldsymbol{x}\mapsto
L(\lambda^{1/\upsilon_{1}}x_{1},\lambda^{1/\upsilon_{2}}x_{2})$ does not
depend on $\lambda>0$. Every generalized invariant function $L$ can be
represented as

$L(\boldsymbol{x})=\tilde{L}(x_{1}/\rho(\boldsymbol{x})^{1/\upsilon_{1}},x_{2}/\rho(\boldsymbol{x})^{1/\upsilon_{2}}),\quad\boldsymbol{x}\in\mathbb{R}^{2}_{0},$
(2.8)

where $\tilde{L}$ is the restriction of $L$ to ${\cal
S}_{1}:=\\{\boldsymbol{x}\in\mathbb{R}^{2}_{0}:\rho(\boldsymbol{x})=1\\}$.

Assumption (F)LRD The spectral density $f$ satisfies

$f(\boldsymbol{x})\ =\
\rho(\boldsymbol{x})^{-1}L(\boldsymbol{x})\big{(}1+o(1)\big{)},\quad|\boldsymbol{x}|\to
0,$ (2.9)

where $\rho(\boldsymbol{x})$ as in (1.4), $\Upsilon>1$ and
$L(\boldsymbol{x})=L(-\boldsymbol{x}),\boldsymbol{x}\in\mathbb{R}^{2}_{0}$ is
a strictly positive continuous generalized invariant function. Moreover, $f$
is bounded on $\\{\boldsymbol{x}\in\Pi^{2}:|\boldsymbol{x}|>\delta\\},$ for
any $\delta>0$.

Assumption (F)LRND,i $(i=1,2)$ The spectral density $f$ satisfies Assumption
(F)LRD except that strict positivity of $L(\boldsymbol{x})$ is replaced by

$\tilde{L}(\boldsymbol{x})=\ell_{i}|x_{i}|^{\mu_{i}}(1+o(1))\qquad(x_{i}\to
0,\,\boldsymbol{x}\in{\cal S}_{1})$ (2.10)

for some $\mu_{i}>0,\ell_{i}>0$.

Assumption (F)ND The spectral density
$f(\boldsymbol{x}),\boldsymbol{x}\in\Pi^{2}$ is a bounded continuous function
such that

$f(\boldsymbol{x})\ =\
\rho(\boldsymbol{x})L(\boldsymbol{x})\big{(}1+o(1)\big{)},\quad|\boldsymbol{x}|\to
0,\quad\text{where}\quad$ (2.11)

where $\rho(\boldsymbol{x})$ as in (1.4), and
$L(\boldsymbol{x}),\boldsymbol{x}\in\mathbb{R}^{2}_{0}$ is as in (2.9).

With generalized homogeneous functions
$\rho(\boldsymbol{x})^{-1}L(\boldsymbol{x}),\rho(\boldsymbol{x})L(\boldsymbol{x})$
in (2.9), (2.11) we associate Gaussian RFs $V_{0,1/\rho},V_{0,\rho}$ by

$\displaystyle
V_{0,1/\rho}(\boldsymbol{x}):=\int_{\mathbb{R}^{2}}\prod_{j=1}^{2}\frac{1-\mathrm{e}^{{\mathrm{i}}u_{j}x_{j}}}{\mathrm{i}u_{j}}\,\sqrt{L(\boldsymbol{u})/\rho(\boldsymbol{u})}\,Z(\mathrm{d}\boldsymbol{u}),\quad
V_{0,\rho}(\boldsymbol{x}):=\int_{\mathbb{R}^{2}}\prod_{j=1}^{2}\frac{1-\mathrm{e}^{{\mathrm{i}}u_{j}x_{j}}}{\mathrm{i}u_{j}}\,\sqrt{L(\boldsymbol{u})\rho(\boldsymbol{u})}\,Z(\mathrm{d}\boldsymbol{u}),$
(2.12)

$\boldsymbol{x}\in\mathbb{R}^{2}_{+},$ where
$\\{Z(\mathrm{d}\boldsymbol{u})\\}$ is a complex-valued Gaussian noise with
zero mean and
$\mathrm{E}|Z(\mathrm{d}\boldsymbol{u})|^{2}=\mathrm{d}\boldsymbol{u}$. The
existence of $V_{0,1/\rho},V_{0,\rho}$ will be established in the following
sections.

######  Definition 2.

Let $V=\\{V(\boldsymbol{x});\boldsymbol{x}\in\mathbb{R}^{2}_{+}\\}$ be a RF
and $\gamma>0,H\geq 0,H_{i}\geq 0,i=1,2$. We say that

(i) $V$ is $(\gamma,H)$-self-similar (SS) if

$\\{V(\lambda
x_{1},\lambda^{\gamma}x_{2});\boldsymbol{x}\in\mathbb{R}^{2}_{+}\\}\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
fdd}}{{=}}\\\
\end{array}\\{\lambda^{H}V(\boldsymbol{x});\boldsymbol{x}\in\mathbb{R}^{2}_{+}\\},\qquad\forall\lambda>0.$
(2.13)

(ii) $V$ is $(H_{1},H_{2})$-multi-self-similar (MSS) is

$\\{V(\lambda_{1}x_{1},\lambda_{2}x_{2});\boldsymbol{x}\in\mathbb{R}^{2}_{+}\\}\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
fdd}}{{=}}\\\
\end{array}\\{\lambda_{1}^{H_{1}}\lambda_{2}^{H_{2}}V(\boldsymbol{x});\boldsymbol{x}\in\mathbb{R}^{2}_{+}\\},\qquad\forall\lambda_{1},\lambda_{2}>0.$
(2.14)

$(\gamma,H)$-SS property is a particular case of the operator self-similarity
property introduced in [2] and corresponding to scaling
$\boldsymbol{x}\to\lambda^{E}\boldsymbol{x}$ with diagonal matrix $E={\rm
diag}(1,\gamma)$. Particularly, $(1,H)$-SS property coincides with the usual
$H$-SS property for RFs on $\mathbb{R}^{2}_{+}$ [25]. $(H_{1},H_{2})$-MSS
property was introduced in [12]. It implies $(\gamma,H)$-SS property with any
$\gamma>0$ and $H=H_{1}+\gamma H_{2}$. Under mild additional assumptions,
scaling limits $V_{\gamma}^{X}$ in (1.2) are $(\gamma,H)$-SS RFs and the
normalization $d_{\lambda,\gamma}$ is regularly varying at infinity with index
$H$ [23].

######  Definition 3.

(Standard) Fractional Brownian Sheet (FBS)
$B_{H_{1},H_{2}}=\\{B_{H_{1},H_{2}}(\boldsymbol{x});\boldsymbol{x}\in\mathbb{R}^{2}_{+}\\}$
with $(H_{1},H_{2})\in[0,1]^{2}$ is defined as a Gaussian process with zero-
mean and covariance function
$\mathrm{E}B_{H_{1},H_{2}}(\boldsymbol{x})B_{H_{1},H_{2}}(\boldsymbol{y})=\prod_{i=1}^{2}r_{H_{i}}(x_{i},y_{i})$,
$\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^{2}_{+}$, where for
$x,y\in\mathbb{R}_{+}$,

$\displaystyle r_{H}(x,y)$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\begin{cases}x^{2H}+y^{2H}-|x-y|^{2H},&0<H\leq 1,\\\
2,&H=0,x=y,\\\ 1,&H=0,x\neq y.\end{cases}$ (2.15)

Usually, FBS is defined for $(H_{1},H_{2})\in(0,1]^{2}$ or even
$(H_{1},H_{2})\in(0,1)^{2}$, see [1]. Extension of FBS to $H_{1}\wedge
H_{2}=0$ was introduced in [28]. (2.15) implies that the restriction of FBS
$B_{H_{1},H_{2}}$ to horizontal/vertical line agrees with fractional Brownian
motion (FBM) $B_{H}=\\{B_{H}(x);x\in\mathbb{R}_{+}\\}$ with covariance
function $r_{H}(x,y)$ and the corresponding Hurst parameter $H=H_{i}\in[0,1]$,
$i=1,2$. Note $r_{0}(x,y)=\lim_{H\downarrow 0}r_{H}(x,y)$. FBM $B_{0}$ is
$0$-SS and an extremely singular (non-measurable) process, see [25,
pp.256–257]. It can be represented as
$B_{0}\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{=}}\\\
\end{array}\\{\frac{1}{\sqrt{2}}(W(x)-W(0));x\in\mathbb{R}_{+}\\}$, where
$W(x)$, $x\in[0,\infty)$, is (uncountable) family of _independent_ $N(0,1)$
r.v.s. The above $B_{0}$ is different from the ’regularized’ FBM with $H=0$
defined in [10, p.2985]. FBS $B_{H_{1},H_{2}}$ with $H_{1}\wedge H_{2}=0$ and
their $\alpha$-stable extensions appeared in limit theorems for RFs [28, 21].
We also recall spectral representation of FBS with
$(H_{1},H_{2})\in(0,1)^{2}$:

$\displaystyle B_{H_{1},H_{2}}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\kappa^{-1}\int_{\mathbb{R}^{2}}\prod_{j=1}^{2}\frac{1-\mathrm{e}^{\mathrm{i}x_{j}u_{j}}}{\mathrm{i}u_{j}|u_{j}|^{H_{j}-\frac{1}{2}}}Z(\mathrm{d}\boldsymbol{u})$
(2.16)

where

$\displaystyle\kappa^{2}$ $\displaystyle:=$
$\displaystyle\int_{\mathbb{R}^{2}}\prod_{j=1}^{2}\frac{|1-\mathrm{e}^{\mathrm{i}u_{j}}|^{2}}{|u_{j}|^{2H_{j}+1}}\mathrm{d}\boldsymbol{u}\
=\ \prod_{j=1}^{2}\frac{\pi}{H_{j}\Gamma(2H_{j})\sin(H_{j}\pi)}$ (2.17)

and $Z(\mathrm{d}\boldsymbol{u})$ is the same Gaussian white noise as in
(2.12); see [1]. For $H_{1}\vee H_{2}=1$, FBS is a line RF of the form
$B_{1,H_{2}}(\boldsymbol{x})=x_{1}B_{H_{2}}(x_{2})$ $(0<H_{2}<1)$,
$B_{H_{1},1}(\boldsymbol{x})=x_{2}B_{H_{1}}(x_{1})$
$(0<H_{1}<1),B_{1,1}(\boldsymbol{x})=x_{1}x_{2}Z$, where $B_{H}$ is FBM and
$Z\sim N(0,1)$. For $H_{1}\wedge H_{2}=0$, FBS allow a construction of finite-
dimensional distributions via independent FBM or uncountable family of
independent Gaussian variables [28].

##  3 Long-range dependence

Define

$\displaystyle H^{+}_{1}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}(1+(\upsilon_{1}\wedge 1)),\quad
H^{+}_{2}:=\frac{1}{2}(1+\upsilon_{2}-\frac{\upsilon_{2}}{\upsilon_{1}\vee
1}),$ (3.1) $\displaystyle H^{-}_{1}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}(1+\upsilon_{1}-\frac{\upsilon_{1}}{\upsilon_{2}\vee
1}),\hskip 14.22636ptH^{-}_{2}:=\frac{1}{2}(1+(\upsilon_{2}\wedge 1)),$

Note $H^{+}_{1}=1$ for $\upsilon_{1}\geq 1$, $H^{+}_{2}=1/2$ for
$\upsilon_{1}\leq 1$. Analogous relations are satisfied by $H^{-}_{i},i=1,2$.

######  Theorem 1.

Let $X$ in (1.1) be a stationary linear RF on $\mathbb{Z}^{2}$ in (1.3) with
spectral density $f$ satisfying Assumption (F)LRD. Then:

  * •

The scaling limits in (1.2) exist for any $\gamma>0$ and satisfy (2.1) with
$\gamma_{0}=\frac{\upsilon_{1}}{\upsilon_{2}}$ and the unbalanced limits are
given by

$\displaystyle V_{+}$ $\displaystyle:=$
$\displaystyle\kappa_{+}\begin{cases}B_{H^{+}_{1},1/2}\,&\upsilon_{1}\leq
1,\\\ B_{1,H^{+}_{2}},&\upsilon_{1}\geq 1,\end{cases}\qquad V_{-}\ :=\
\kappa_{-}\begin{cases}B_{1/2,H^{-}_{2}}&\upsilon_{2}\leq 1,\\\
B_{H^{-}_{1},1},&\upsilon_{2}\geq 1,\end{cases}$ (3.2)

where $H^{\pm}_{i},i=1,2$ as in (3.1) and the constants $\kappa_{\pm}>0$ in
(3.2) are defined in (3.13), (3.14) (or can be determined from these
expressions by symmetry).

  * •

The well-balanced limit $V_{0}=V_{0,1/\rho}$ is given in (2.12).

  * •

The normalization in (1.2) is given by
$d_{\lambda,\gamma}:=\lambda^{H(\gamma)}(\log_{+}\lambda)^{1/2}$ if
$\gamma>\gamma_{0}$ and $\upsilon_{1}=1$ or $\gamma<\gamma_{0}$ and
$\upsilon_{2}=1$, and $d_{\lambda,\gamma}:=\lambda^{H(\gamma)}$ in the
remaining cases, where $H(\gamma)$ is the weighted linear combination with
weights $(1,\gamma)$ of the Hurst indices of FBM on the r.h.s. (3.2), viz.,

$\displaystyle H(\gamma)=\begin{cases}H^{+}_{1}+\gamma
H^{+}_{2},&\gamma\geq\gamma_{0},\\\ H^{-}_{1}+\gamma
H^{-}_{2},&\gamma\leq\gamma_{0}.\end{cases}$ (3.3)

  * •

The RF $X$ exhibits scaling transition at $\gamma_{0}$.

Proof. The simple bound $\big{|}\sum_{\boldsymbol{t}\in K_{[\lambda
x_{1},\lambda^{\gamma}x_{2}]}}a(\boldsymbol{t}-\boldsymbol{u})\big{|}\leq
C\lambda^{(1+\gamma)/2}\big{(}\sum_{\boldsymbol{t}\in\mathbb{Z}^{2}}a(\boldsymbol{t})^{2}\big{)}^{1/2}\leq
C\lambda^{(1+\gamma)/2}$ reduces the criterion (2.4) to
$H(\gamma)>(1+\gamma)/2$. This follows from (3.3) since $H^{\pm}_{i}\geq
1/2,i=1,2$. Whence, for (1.2) it suffices to prove the convergence of
covariance functions in (2.3). The latter relation is essentially proved in
[24] except for the cases $\upsilon_{1}=1$ or $\upsilon_{2}=1$. The subsequent
proof is limited to these cases. Note that the definitions in (3.2), (3.3)
given by different expressions agree for $\upsilon_{i}=1,i=1,2$ and/or
$\gamma=\gamma_{0}$. By symmetry, it suffices to discuss the case
$\upsilon_{1}=1$ only and consider the limits in (2.3) for
$\gamma>\gamma_{0},\gamma<\gamma_{0},$ and $\gamma=\gamma_{0}$ separately.
W.l.g. $\lambda>1$.

Case $\gamma>\gamma_{0}=1/\upsilon_{2}\,(\upsilon_{1}=1)$. In this case,
$V_{+}=\kappa_{+}B_{1,1/2}$ and
$d^{2}_{\lambda,\gamma}=\lambda^{2+\gamma}\log\lambda$, according to (3.2). In
the integral in (2.2), change the variables: $u_{1}\to
u_{1}/\lambda^{\gamma\upsilon_{2}},u_{2}\to u_{2}/\lambda^{\gamma}$ and note
that

$\displaystyle\lambda^{-1}D_{[\lambda
x_{1}]}(u_{1}/\lambda^{\gamma\upsilon_{2}})\to
x_{1},\qquad\lambda^{-\gamma}D_{[\lambda^{\gamma}x_{2}]}(u_{2}/\lambda^{\gamma})\to\frac{1-\mathrm{e}^{\mathrm{i}x_{2}u_{2}}}{-\mathrm{i}u_{2}},$
(3.4) $\displaystyle
f_{\lambda}(\boldsymbol{u}):=\lambda^{-\gamma\upsilon_{2}}f(u_{1}/\lambda^{\gamma\upsilon_{2}},u_{2}/\lambda^{\gamma})\to
L(\boldsymbol{u})/\rho(\boldsymbol{u})=:f_{0}(\boldsymbol{u})$

point-wise for any $\boldsymbol{u}\in\mathbb{R}^{2}_{0}$ as
$\lambda\to\infty$, due to $\gamma>\gamma_{0}$. Moreover, (2.9) implies

$|f_{\lambda}(\boldsymbol{u})-f_{0}(\boldsymbol{u})|\leq\frac{1}{|u_{1}|+|u_{2}|^{\upsilon_{2}}}\delta\big{(}\frac{u_{1}}{\lambda^{\gamma\upsilon_{2}}},\frac{u_{2}}{\lambda^{\gamma}}\big{)},$
(3.5)

where $\delta(\boldsymbol{u})$ is a bounded function tending to 0 as
$\boldsymbol{u}\to{\bf 0}$. Thus,
$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})=\sum_{i=1}^{2}R^{(i)}_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})$,
where
$R^{(i)}_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y}):=\frac{1}{\log\lambda}\int_{\lambda^{\gamma\upsilon_{2}}\Pi\times\lambda^{\gamma}\Pi}G^{(i)}_{\lambda}(\boldsymbol{u})\mathrm{d}\boldsymbol{u},i=1,2$
with

$\displaystyle G^{(1)}_{\lambda}(\boldsymbol{u})$ $\displaystyle:=$
$\displaystyle\frac{D_{[\lambda
x_{1}]}(u_{1}/\lambda^{\gamma\upsilon_{2}})\overline{D_{[\lambda
y_{1}]}(u_{1}/\lambda^{\gamma\upsilon_{2}})}}{\lambda^{2}}\cdot\frac{D_{[\lambda^{\gamma}x_{2}]}(u_{2}/\lambda^{\gamma})\overline{D_{[\lambda^{\gamma}y_{2}]}(u_{2}/\lambda^{\gamma})}}{\lambda^{2\gamma}}\cdot
f_{0}(\boldsymbol{u})$ $\displaystyle\to$ $\displaystyle
x_{1}y_{1}\cdot\frac{(1-\mathrm{e}^{\mathrm{i}x_{2}u_{2}})(1-\mathrm{e}^{-\mathrm{i}y_{2}u_{2}})}{|u_{2}|^{2}}\cdot
f_{0}(\boldsymbol{u})=:G(\boldsymbol{u}),$

and $G^{(2)}_{\lambda}(\boldsymbol{u})$ analogously defined with
$f_{0}(\boldsymbol{u})$ replaced by
$f_{\lambda}(\boldsymbol{u})-f_{0}(\boldsymbol{u})$.

Let us prove $R^{(2)}_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})\to 0$.
From elementary bound $|D_{n}(x)|\leq Cn/(1+n|x|),x\in\Pi$ and (3.5) we get

$\displaystyle|R^{(2)}_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})|$
$\displaystyle\leq$
$\displaystyle\frac{C}{\log\lambda}\int_{\lambda^{\gamma\upsilon_{2}}\Pi\times\lambda^{\gamma}\Pi}\frac{\mathrm{d}\boldsymbol{u}}{(1+u_{2}^{2})(|u_{1}|+|u_{2}|^{\upsilon_{2}})}\delta\big{(}\frac{u_{1}}{\lambda^{\gamma\upsilon_{2}}},\frac{u_{2}}{\lambda^{\gamma}}\big{)}$
(3.7) $\displaystyle=$
$\displaystyle\frac{C}{\log\lambda}\int_{\mathbb{R}}\frac{\mathrm{d}u_{2}}{1+u_{2}^{2}}\int_{|u_{1}|\leq\lambda^{\gamma\upsilon_{2}}/|u_{2}|^{\upsilon_{2}}}\frac{\mathrm{d}u_{1}}{|u_{1}|+1}\delta\big{(}\frac{u_{1}|u_{2}|^{\upsilon_{2}}}{\lambda^{\gamma\upsilon_{2}}},\frac{u_{2}}{\lambda^{\gamma}}\big{)}.$

Split the integration region on the r.h.s. of (3.7) into three parts
corresponding to $\\{\frac{|u_{2}|}{\lambda^{\gamma}}\geq\epsilon\\}$,
$\\{\frac{|u_{2}|}{\lambda^{\gamma}}<\epsilon,|u_{1}|(\frac{|u_{2}|}{\lambda^{\gamma}})^{\upsilon_{2}}<\epsilon\\}$,
and
$\\{\frac{|u_{2}|}{\lambda^{\gamma}}<\epsilon,|u_{1}|(\frac{|u_{2}|}{\lambda^{\gamma}})^{\upsilon_{2}}\geq\epsilon\\}$,
$T_{1},T_{2}$ and $T_{3}$, say, so that
$|R^{(2)}_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})|\leq
C(\log\lambda)^{-1}\sum_{i=1}^{3}\int_{T_{i}}\dots$
$=:(\log\lambda)^{-1}\sum_{i=1}^{3}R_{i}$. The inner integral in (3.7) on
$T_{1}$ is bounded by $C\log(1/\epsilon)^{\upsilon_{2}})$ implying $R_{1}\to
0\ (\lambda\to\infty,\,\forall\epsilon>0)$. Next, by the property
$\delta(\boldsymbol{u})\to 0$, for any $\delta^{\prime}>0$ there exists
$\epsilon>0$ s.t.
$R_{2}\leq\delta^{\prime}(\log\lambda)^{-1}\int_{\mathbb{R}}(1+u^{2}_{2})^{-1}\log(\lambda^{\gamma}/|u_{2}|)^{\upsilon_{2}}\mathrm{d}u_{2}\leq
C\delta^{\prime}$ vanishes with $\epsilon\to 0$ uniformly in $\lambda\geq 2$.
Finally, for any $\epsilon>0$ fixed, $R_{3}\leq
C(\log\lambda)^{-1}\int_{\mathbb{R}}(1+u^{2}_{2})^{-1}\big{\\{}\log(\lambda^{\gamma}/|u_{2}|)^{\upsilon_{2}}-\log\epsilon(\lambda^{\gamma}/|u_{2}|)^{\upsilon_{2}}\big{\\}}=O(1/\log\lambda)\to
0$ as $\lambda\to\infty$, thus ending the proof of
$R^{(2)}_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})\to 0$.

Next, we prove the limits

$\lim_{\lambda\to\infty}R^{(1)}_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})=\lim_{\lambda\to\infty}\tilde{R}_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})=\kappa_{+}^{2}\mathrm{E}[B_{1,1/2}(\boldsymbol{x})B_{1,1/2}(\boldsymbol{y})]=\kappa_{+}^{2}x_{1}y_{1}(x_{2}\wedge
y_{2})$ (3.8)

where

$\displaystyle\tilde{R}_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle:=$
$\displaystyle\frac{1}{\log\lambda}\int_{\lambda^{\gamma\upsilon_{2}}\Pi\times\lambda^{\gamma}\Pi}G(\boldsymbol{u})\mathrm{d}\boldsymbol{u}$
(3.9) $\displaystyle=$ $\displaystyle
x_{1}y_{1}\int_{|u_{2}|\leq\lambda^{\gamma}\pi}\frac{(1-\mathrm{e}^{\mathrm{i}x_{2}u_{2}})(1-\mathrm{e}^{-\mathrm{i}y_{2}u_{2}})}{|u_{2}|^{2}}\mathrm{d}u_{2}\times\frac{1}{\log\lambda}\int_{|u_{1}|\leq\lambda^{\gamma\upsilon_{2}}\pi}f_{0}(\boldsymbol{u})\mathrm{d}u_{1}.$

Consider the second limit in (3.8). By definition,
$f_{0}(\boldsymbol{u})=L\big{(}\frac{u_{1}}{|u_{1}|+|u_{2}|^{\upsilon_{2}}},\frac{u_{2}}{(|u_{1}|+|u_{2}|^{\upsilon_{2}})^{1/\upsilon_{2}}}\big{)}(|u_{1}|+|u_{2}|^{\upsilon_{2}})^{-1}$
behaves as $L(1,0)|u_{1}|^{-1}$ when $|u_{1}|\to\infty$, indeed, for any
$u_{2}\neq 0$

$|u_{2}|^{\upsilon_{2}}(|u_{1}|+1)f_{0}(u_{1}|u_{2}|^{\upsilon_{2}},u_{2})=L\Big{(}\frac{u_{1}+1}{|u_{1}|+1},\frac{{\rm
sgn}(u_{2})}{|u_{1}|+1}\Big{)}\ \to\ L(1,0),\qquad|u_{1}|\to\infty$ (3.10)

by the conditions on $L(\cdot)$ in the theorem. Then, the last term on the
r.h.s. of (3.9) can be rewritten as the sum of two terms: the first term
$\frac{L(1,0)}{\log\lambda}\int_{|u_{1}|\leq(\lambda^{\gamma}/|u_{2}|)^{\upsilon_{2}}\pi}(|u_{1}|+1)^{-1}\mathrm{d}u_{1}\to
2L(1,0)\gamma\upsilon_{2}\ (\forall u_{2}\neq 0)$, and the second term

$\frac{1}{\log\lambda}\int_{|u_{1}|\leq(\lambda^{\gamma}/|u_{2}|)^{\upsilon_{2}}\pi}\frac{\mathrm{d}u_{1}}{|u_{1}|+1}\Big{(}L\big{(}\frac{u_{1}+1}{|u_{1}|+1},\frac{{\rm
sgn}(u_{2})}{|u_{1}|+1}\big{)}-L(1,0)\Big{)}\ \to\ 0\qquad(\forall\,u_{2}\neq
0)$ (3.11)

in view of (3.10) and boundedness of $L(\cdot)$. The proof of the second limit
in (3.8) follows from (3.9)-(3.11) and the DCT using the bound
$(\log\lambda)^{-1}\big{|}\int_{|u_{1}|\leq\lambda^{\gamma\upsilon_{2}}\pi}g_{+}(\boldsymbol{u})\mathrm{d}u_{1}\big{|}\leq
C(\log\lambda)^{-1}\int_{|u_{1}|\leq\lambda^{\gamma\upsilon_{2}}\pi}(|u_{1}|+|u_{2}|^{\upsilon_{2}})^{-1}\mathrm{d}u_{1}\leq
C(1+|\log(|u_{2}|)|)$ and the integrability of
$(1+|\log(|u_{2}|)|)(1+u_{2}^{2})^{-1}$ on $\mathbb{R}$.

It remains to prove that
$R^{(1)}_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})-\tilde{R}_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})\to
0$. W.l.g., take $\boldsymbol{x}=\boldsymbol{y}={\bf 1}$ and let
$R^{(1)}_{\lambda,\gamma}({\bf 1},{\bf 1})-\tilde{R}_{\lambda,\gamma}({\bf
1},{\bf 1})=:R^{\prime}_{\lambda,\gamma}$. Then

$\displaystyle|R^{\prime}_{\lambda,\gamma}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{\log\lambda}\int_{\lambda^{\gamma\upsilon_{2}}\Pi\times\lambda^{\gamma}\Pi}\Big{|}\frac{|D_{[\lambda]}(u_{1}/\lambda^{\gamma\upsilon_{2}})D_{[\lambda^{\gamma}]}(u_{2}/\lambda^{\gamma})|^{2}}{\lambda^{2+2\gamma}}-\frac{|1-\mathrm{e}^{\mathrm{i}u_{2}}|^{2}}{u_{2}^{2}}\Big{|}\frac{\mathrm{d}\boldsymbol{u}}{|u_{1}|+|u_{2}|^{\upsilon_{2}}}$
(3.12) $\displaystyle=$
$\displaystyle\frac{1}{\log\lambda}\int_{\lambda^{\gamma\upsilon_{2}}\Pi\times\lambda^{\gamma}\Pi}\frac{\delta_{\lambda}(\boldsymbol{u})\mathrm{d}\boldsymbol{u}}{(1+u_{2}^{2})(|u_{1}|+|u_{2}|^{\upsilon_{2}})}\
=\ o(1)$

since $\delta_{\lambda}(\boldsymbol{u})\to 0$ uniformly in $\boldsymbol{u}$ in
the last integral, see (4.6), and the r.h.s. of (3.12) with
$\delta_{\lambda}(\boldsymbol{u})$ replaced by 1 is bounded. This ends the
proof of (3.8) with

$\kappa_{+}^{2}:=2L(1,0)\gamma\upsilon_{2}\int_{\mathbb{R}}|(1-\mathrm{e}^{\mathrm{i}u})/u|^{2}\mathrm{d}u\
=\ 4\pi L(1,0)\gamma\upsilon_{2}$ (3.13)

and completes the proof of (2.3) when $\gamma>\gamma_{0}$.

Case $\gamma<\gamma_{0}=1/\upsilon_{2}\,(\upsilon_{1}=1)$. There are three
subcases: (a) $\upsilon_{2}<1,$ (b) $\upsilon_{2}>1$, and (c)
$\upsilon_{2}=1$. Accordingly, $V_{-}=\kappa_{-}B_{1/2,H^{-}_{2}}$ in subcase
(a), $V_{-}=\kappa_{-}B_{H^{-}_{1},1}$ in subcase (b), and
$V_{-}=\kappa_{-}B_{1/2,1}$ in subcase (c). Subcases (a) and (b) do not
involve logarithmic normalization and are essentially treated in [24,
Thm.3.1]. Subcase (c) is symmetric to the above case
$\gamma>\gamma^{X}_{0},\upsilon_{1}=1$ with $(x_{1},\upsilon_{1})$ and
$(x_{2},\upsilon_{2})$ exchanged, by noting that the latter discussion applies
to any $\upsilon_{2}>0$ including $\upsilon_{2}=1$.

Case $\gamma=\gamma_{0}=1/\upsilon_{2}\,(\upsilon_{1}=1)$. In this case,
$V_{0}=V_{0,1/\rho}$ is given in (2.12) and the result follows from [24] with
small changes, thus ending the proof of Theorem 1. $\Box$

######  Remark 1.

The squared asymptotic constant in (3.2) for $\upsilon_{1}\neq 1$ takes the
form

$\displaystyle\kappa^{2}_{+}$ $\displaystyle=$
$\displaystyle\begin{cases}L(1,0)\int_{\mathbb{R}^{2}}\prod_{j=1}^{2}\big{|}(1-\mathrm{e}^{\mathrm{i}u_{j}})/u_{j}\big{|}^{2}|u_{1}|^{-\upsilon_{1}}\mathrm{d}\boldsymbol{u},&\upsilon_{1}<1,\\\
\int_{\mathbb{R}^{2}}\big{|}(1-\mathrm{e}^{\mathrm{i}u_{2}})/u_{2}\big{|}^{2}L(\boldsymbol{u})\rho(\boldsymbol{u})^{-1}\mathrm{d}\boldsymbol{u},&\upsilon_{1}>1,\end{cases}$
(3.14)

($\kappa^{2}_{-}$ is defined symmetrically with
$u_{1},u_{2},\upsilon_{1},\upsilon_{2}$ exchanged by
$u_{2},u_{1},\upsilon_{2},\upsilon_{1}$). For $\upsilon_{1}<1$ the integral in
(3.14) can be explicitly evaluated as
$\kappa^{2}_{+}=L(1,0)(2\pi)^{2}/\Gamma(2+\upsilon_{1})\sin((1+\upsilon_{1})\pi/2),$
see (2.17).

##  4 Negative dependence

This sec. describes anisotropic scaling limits in (1.2) of linear RFs
satisfying Assumption (F)ND. Define

$\displaystyle H^{+}_{1}:=\frac{1}{2}(1-(\upsilon_{1}\wedge 1)),\qquad
H^{-}_{2}:=\frac{1}{2}(1-(\upsilon_{2}\wedge
1)),\qquad\gamma_{0}:=\frac{\upsilon_{1}\wedge 1}{\upsilon_{2}\wedge 1}.$
(4.1)

Note $H^{+}_{1},H^{-}_{2}\in[0,1/2)$ and $H^{+}_{1}=0$ (respectively,
$H^{-}_{2}=0$) is equivalent to $\upsilon_{1}\geq 1$ (respectively,
$\upsilon_{2}\geq 1$). We also set $H^{+}_{2}=H^{-}_{1}:=1/2$.

######  Theorem 2.

Let $X$ in (1.1) be a stationary linear RF on $\mathbb{Z}^{2}$ in (1.3) with
spectral density $f$ satisfying Assumption (F)ND. Then:

  * •

The scaling limits in (1.2) exist for any $\gamma>0$ and satisfy (2.1) with
$\gamma_{0},H^{\pm}_{i}$ in (4.1) and the unbalanced limits given by

$\displaystyle V_{+}$ $\displaystyle:=$
$\displaystyle\kappa_{+}B_{H^{+}_{1},1/2},\qquad V_{-}\ :=\
\kappa_{-}B_{1/2,H^{-}_{2}}.$ (4.2)

The asymptotic constants $\kappa_{\pm}>0$ are written in (4.9), (4.14),
(4.19), (4.21), and (LABEL:kappa6) (or can be determined from these
expressions by symmetry).

  * •

The well-balanced limit is given by

$\displaystyle V_{0}$ $\displaystyle:=$
$\displaystyle\begin{cases}V_{0,\rho},&H^{+}_{1}\wedge H^{-}_{2}>0,\\\
\kappa_{+}B_{H^{+}_{1},1/2}+\kappa_{-}B_{1/2,H^{-}_{2}},&H^{+}_{1}\wedge
H^{-}_{2}=0,\end{cases}$ (4.3)

where $B_{H^{+}_{1},1/2}$ and $B_{1/2,H^{-}_{2}}$ are independent and
$V_{0,\rho}$ is defined in (2.12).

  * •

The normalization in (1.2) is given by
$d_{\lambda,\gamma}:=\lambda^{H(\gamma)}(\log_{+}\lambda)^{1/2}$ in the cases
$\gamma\geq\gamma_{0},H^{+}_{1}=0$ or $\gamma\leq\gamma_{0},H^{-}_{2}=0$, and
$d_{\lambda,\gamma}:=\lambda^{H(\gamma)}$ in the remaining cases, with

$\displaystyle H(\gamma)$ $\displaystyle:=$
$\displaystyle\begin{cases}H^{+}_{1}+(\gamma/2),&\gamma\geq\gamma_{0},\\\
(1/2)+\gamma H^{-}_{2},&\gamma\leq\gamma_{0}.\end{cases}$ (4.4)

  * •

The RF $X$ exhibits scaling transition at $\gamma_{0}$.

Proof. Similarly as in the proof of Theorem 1 we check the asymptotic
gaussianity criterion in (2.4), reducing the proof to the convergence in (2.3)
of covariance functions. Relation
$f(\boldsymbol{u})=(2\pi)^{2}|\widehat{a}(\boldsymbol{u})|^{2}$ and the
assumptions on $f$ imply that $|\widehat{a}(\boldsymbol{u})|\leq C$ is
bounded. Whence, with $(\gamma_{1},\gamma_{2}):=(1,\gamma)$, we get that
$\sup_{\boldsymbol{s}\in\mathbb{Z}^{2}}\big{|}\sum_{\boldsymbol{t}\in
K_{[\lambda
x_{1},\lambda^{\gamma}x_{2}]}}a(\boldsymbol{t}-\boldsymbol{s})\big{|}\leq\int_{\Pi^{2}}\prod_{i=1}^{2}\big{|}D_{[\lambda^{\gamma_{i}}x_{i}]}(u_{i})\big{|}\,|\widehat{a}(\boldsymbol{u})|\mathrm{d}\boldsymbol{u}\leq
C\int_{\Pi^{2}}\prod_{i=1}^{2}\big{|}D_{[\lambda^{\gamma_{i}}x_{i}]}(u_{i})\big{|}\,\mathrm{d}\boldsymbol{u}\leq
C(\log\lambda)^{2}$ so that (2.4) holds since $H(\gamma)$ in (4.4) satisfy
$H(\gamma)>0$.

The subsequent proof of (2.3) is split into parts I and II according to
whether $\upsilon_{i}\neq 1,i=1,2,$ or $\upsilon_{i}=1\,(\exists i=1,2)$. In
turn, each part is split into several cases and subcases which are treated
separately. Part II involves the logarithmic factor in the normalization and
is more delicate.

Part I. Case
$\upsilon_{1}\vee\upsilon_{2}<1,\gamma_{0}=\upsilon_{1}/\upsilon_{2}$.  Then
$H^{+}_{1}=(1-\upsilon_{1})/2\in(0,1/2),H^{-}_{2}=(1-\upsilon_{2})/2\in(0,1/2)$.
By change of variables: $u_{1}\to u_{1}/\lambda,u_{2}\to
u_{2}/\lambda^{\gamma}$ we rewrite (2.2) as
$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})=\int_{\lambda\Pi\times\lambda^{\gamma}\Pi}G_{\lambda}(\boldsymbol{u})\mathrm{d}\boldsymbol{u}$
where

$\displaystyle G_{\lambda}(\boldsymbol{u})$ $\displaystyle:=$
$\displaystyle\frac{D_{[\lambda x_{1}]}(u_{1}/\lambda)\overline{D_{[\lambda
y_{1}]}(u_{1}/\lambda)}}{\lambda^{2}}\cdot\frac{D_{[\lambda^{\gamma}x_{2}]}(u_{2}/\lambda^{\gamma})\overline{D_{[\lambda^{\gamma}y_{2}]}(u_{2}/\lambda^{\gamma})}}{\lambda^{2\gamma}}\cdot
f_{\lambda}(\boldsymbol{u}),$ (4.5) $\displaystyle
f_{\lambda}(\boldsymbol{u})$ $\displaystyle:=$ $\displaystyle
f(u_{1}/\lambda,u_{2}/\lambda^{\gamma})/\lambda^{2H(\gamma)-1-\gamma}.$

By (2.9) and the definition of $H(\gamma)$ in (4.4),

$\displaystyle\lambda^{-1}D_{[\lambda
x_{1}]}(u_{1}/\lambda)\to\frac{1-\mathrm{e}^{\mathrm{i}x_{1}u_{1}}}{-\mathrm{i}u_{1}},\qquad\lambda^{-\gamma}D_{[\lambda^{\gamma}x_{2}]}(u_{2}/\lambda^{\gamma})\to\frac{1-\mathrm{e}^{\mathrm{i}x_{2}u_{2}}}{-\mathrm{i}u_{2}},$
(4.6) $\displaystyle f_{\lambda}(\boldsymbol{u})\to
g_{\gamma}(\boldsymbol{u})\ :=\
\begin{cases}f_{0}(\boldsymbol{u}),&\gamma=\gamma_{0},\\\
f_{0}(u_{1},0),&\gamma>\gamma_{0},\\\
f_{0}(0,u_{2}),&\gamma<\gamma_{0},\end{cases}$

point-wise for any $\boldsymbol{u}\in\mathbb{R}^{2}_{0}$, where
$f_{0}(\boldsymbol{u}):=L(\boldsymbol{u})\rho(\boldsymbol{u})$, see (2.11).
Particularly, $f_{0}(u_{1},0)=|u_{1}|^{-\upsilon_{1}}L(1,0)$,
$f_{0}(0,u_{2})=|u_{2}|^{-\upsilon_{2}}L(0,1),\boldsymbol{u}=(u_{1},u_{2})\in\mathbb{R}^{2}_{0}$
since $L(1,0)=L(-1,0),$ $L(0,1)=L(0,-1)$. We also see from (2.16), (2.12) that
the covariance function of the limit RF with appropriately chosen
$\kappa_{\pm}$ writes as
$\mathrm{E}V_{\gamma}(\boldsymbol{x})V_{\gamma}(\boldsymbol{y})=\int_{\mathbb{R}^{2}}G(\boldsymbol{u})\mathrm{d}\boldsymbol{u}$,
where

$G(\boldsymbol{u})\ :=\
\prod_{j=1}^{2}\big{(}\frac{1-\mathrm{e}^{\mathrm{i}x_{j}u_{j}}}{\mathrm{i}u_{j}}\big{)}\big{(}\frac{1-\mathrm{e}^{-\mathrm{i}y_{j}u_{j}}}{-\mathrm{i}u_{j}}\big{)}g_{\gamma}(\boldsymbol{u})$
(4.7)

is the product of the corresponding limit functions in (4.6) and
$G_{\lambda}(\boldsymbol{u})\to
G(\boldsymbol{u}),\boldsymbol{u}\in\mathbb{R}^{2}_{0}$ according to (4.6). The
proof of
$\int_{\mathbb{R}^{2}}G_{\lambda}(\boldsymbol{u})\mathrm{d}\boldsymbol{u}\to\int_{\mathbb{R}^{2}}G(\boldsymbol{u})\mathrm{d}\boldsymbol{u}$
in all three cases $\gamma>\gamma_{0},\gamma<\gamma_{0}$, and
$\gamma=\gamma_{0}$ now follows from the dominating convergence theorem (DCT)
using the bound

$|\lambda^{-1}D_{[\lambda x]}(\mbox{$\frac{u}{\lambda}$})|\leq Cx(1+|([\lambda
x]/\lambda)u|)\leq C/(1+|u|),\qquad|u|<\lambda\pi,$

for any fixed $x\in\mathbb{R},x\neq 0$, implying

$\displaystyle|G_{\lambda}(\boldsymbol{u})|\ \leq\
C\prod_{i=1}^{2}(1+u_{i}^{2})^{-1}\times\begin{cases}\rho(\boldsymbol{u}),&\gamma=\gamma_{0},\\\
\rho(u_{1},0),&\gamma>\gamma_{0},\\\
\rho(0,u_{2}),&\gamma<\gamma_{0}.\end{cases}$ (4.8)

Hence, the r.h.s. of (4.8) denoted by $\bar{G}(\boldsymbol{u})$ is integrable:
$\int_{\mathbb{R}^{2}}\bar{G}(\boldsymbol{u})\mathrm{d}\boldsymbol{u}<\infty$.
The asymptotic constants $\kappa_{\pm}$ in (4.2) are determined by
$\int_{\mathbb{R}^{2}}|G(\boldsymbol{u})|^{2}\mathrm{d}\boldsymbol{u}=1/\kappa^{2}_{+}(\gamma>\gamma_{0}),=1/\kappa^{2}_{-}(\gamma<\gamma_{0})$
for $\boldsymbol{x}=\boldsymbol{y}={\bf 1}$ in (4.7) and take a similar form
as in (3.14); particularly,

$\displaystyle\kappa_{+}^{2}$ $\displaystyle:=$ $\displaystyle
L(1,0)\int_{\mathbb{R}^{2}}|u_{1}|^{\upsilon_{1}}\prod_{j=1}^{2}|(1-\mathrm{e}^{{\mathrm{i}}u_{j}})/u_{j}|^{2}\mathrm{d}\boldsymbol{u}.$
(4.9)

Case $\upsilon_{1}<1<\upsilon_{2},\gamma_{0}=\upsilon_{1}$.

Subcase $\gamma>\upsilon_{1}$. We prove (2.3) with the same
$V_{+}=\kappa_{+}B_{H^{+}_{1},1/2}$ as in the previous case using a modified
argument, as follows. Split
$f(\boldsymbol{u})=f_{1}(\boldsymbol{u})+\tilde{f}_{1}(\boldsymbol{u}),$ where
$f_{1}(\boldsymbol{u})=f(u_{1},0),\tilde{f}_{1}(\boldsymbol{u}):=f(\boldsymbol{u})-f(u_{1},0)$
and accordingly,
$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})=R_{1}(\boldsymbol{x},\boldsymbol{y})+\tilde{R}_{1}(\boldsymbol{x},\boldsymbol{y})$.
The convergence
$R_{1}(\boldsymbol{x},\boldsymbol{y})\to\kappa^{2}_{+}\mathrm{E}B_{H^{+}_{1},1/2}(\boldsymbol{x})B_{H^{+}_{1},1/2}(\boldsymbol{y})$
for any $\gamma>0$ follows as in the case $\upsilon_{1}\vee\upsilon_{2}<1$
above. Whence, it suffices to prove
$\tilde{R}_{1}(\boldsymbol{x},\boldsymbol{x})\to 0$, or

$\displaystyle
J_{\lambda}:=\int_{\Pi^{2}}|\tilde{f}_{1}(\boldsymbol{u})|\,|D_{[\lambda
x_{1}]}(u_{1})D_{[\lambda^{\gamma}x_{2}]}(u_{2})|^{2}\mathrm{d}\boldsymbol{u}$
$\displaystyle=$ $\displaystyle o(\lambda^{1-\upsilon_{1}+\gamma}).$ (4.10)

We have $|\tilde{f}_{1}(\boldsymbol{u})|\leq
C\sum_{k=1}^{3}|h_{1,k}(\boldsymbol{u})|$, where
$h_{1,1}(\boldsymbol{u}):=\rho(\boldsymbol{u})-\rho(u_{1},0)=|u_{2}|^{\upsilon_{2}},h_{1,2}(\boldsymbol{u}):=\rho(u_{1},0)(L(\boldsymbol{u})-L(u_{1},0))$
and $h_{1,3}(\boldsymbol{u})=\rho(\boldsymbol{u})\delta(\boldsymbol{u})$,
where $\delta(\boldsymbol{u})$ is a bounded function tending to $0$ as
$|\boldsymbol{u}|\to 0$. Accordingly, $J_{\lambda}\leq
C\sum_{k=1}^{3}J_{\lambda,k}$ where $J_{\lambda,k}$ is defined as in (4.10)
with $|\tilde{f}_{1}(\boldsymbol{u})|$ replaced by
$|h_{1,k}(\boldsymbol{u})|$. Here,

$\displaystyle J_{\lambda,1}$ $\displaystyle\leq$
$\displaystyle\int_{\Pi^{2}}|u_{2}|^{\upsilon_{2}}\frac{|1-\mathrm{e}^{\mathrm{i}[\lambda
x_{1}]u_{1}}|^{2}}{|u_{1}|^{2}|u_{2}|^{2}}\mathrm{d}\boldsymbol{u}\
\leq\lambda\int_{\mathbb{R}\times\Pi}\frac{|1-\mathrm{e}^{\mathrm{i}u_{1}x_{1}}|^{2}}{|u_{1}|^{2}}|u_{2}|^{\upsilon_{2}-2}\mathrm{d}\boldsymbol{u}\leq
C\lambda=o(\lambda^{1-\upsilon_{1}+\gamma})$ (4.11)

since $\gamma>\upsilon_{1}$. Next,

$\displaystyle J_{\lambda,2}$ $\displaystyle\leq$
$\displaystyle\int_{\Pi}|u_{1}|^{\upsilon_{1}}\frac{|1-\mathrm{e}^{\mathrm{i}[\lambda
x_{1}]u_{1}}|^{2}}{|u_{1}|^{2}}\mathrm{d}u_{1}\times\int_{\Pi}|L(u_{1},u_{2})-L(u_{1},0)|\frac{|1-\mathrm{e}^{\mathrm{i}[\lambda^{\gamma}x_{2}]u_{2}}|^{2}}{|u_{2}|^{2}}\mathrm{d}u_{2}$
$\displaystyle\leq$ $\displaystyle
C\lambda^{1-\upsilon_{1}+\gamma}\int_{\mathbb{R}}|u_{1}|^{\upsilon_{1}}\frac{|1-\mathrm{e}^{\mathrm{i}u_{1}x_{1}}|^{2}}{|u_{1}|^{2}}\mathrm{d}u_{1}\times\int_{\mathbb{R}}\big{|}L\big{(}\frac{u_{1}}{\lambda},\frac{u_{2}}{\lambda^{\gamma}}\big{)}-L\big{(}\frac{u_{1}}{\lambda},0\big{)}\big{|}\frac{|1-\mathrm{e}^{\mathrm{i}u_{2}x_{2}}|^{2}}{|u_{2}|^{2}}\mathrm{d}u_{2}$
$\displaystyle=$ $\displaystyle o(\lambda^{1-\upsilon_{1}+\gamma})$

by the DCT and the fact that
$\big{|}L\big{(}\frac{u_{1}}{\lambda},\frac{u_{2}}{\lambda^{\gamma}}\big{)}-L\big{(}\frac{u_{1}}{\lambda},0\big{)}\big{|}$
is bounded and tends to 0 for any $\boldsymbol{u}\in\mathbb{R}^{2}_{0}$ by the
continuity of $\tilde{L}$ in (2.8) and the fact that
$\gamma>\upsilon_{1}>\upsilon_{1}/\upsilon_{2}$. A similar argument also
entails $J_{\lambda,3}=o(\lambda^{1-\upsilon_{1}+\gamma})$, proving (4.10) and
the convergence (1.2) in the subcase $\gamma>\upsilon_{1}$.

Subcase $\gamma<\upsilon_{1}$. We prove (2.3) with
$V_{-}=\kappa_{-}B_{1/2,0}$, $d^{2}_{\lambda,\gamma}=\lambda$ and $\kappa_{-}$
in (4.14). Split
$f(\boldsymbol{u})=f_{2}(\boldsymbol{u})+\tilde{f}_{2}(\boldsymbol{u}),\tilde{f}_{2}(\boldsymbol{u}):=f(\boldsymbol{u})-f(0,u_{2})$
and accordingly,
$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})=R_{2}(\boldsymbol{x},\boldsymbol{y})+\tilde{R}_{2}(\boldsymbol{x},\boldsymbol{y})$.
It suffices to prove

$\displaystyle
R_{2}(\boldsymbol{x},\boldsymbol{y})\to\kappa_{-}^{2}(x_{1}\wedge
y_{1})\times\frac{1}{2}(1+\mathbb{I}(x_{2}=y_{2}))\quad\text{and}\quad\tilde{R}_{2}(\boldsymbol{x},\boldsymbol{x})\to
0.$ (4.13)

To show the first relation in (4.13), note that

$\displaystyle R_{2}(\boldsymbol{x},\boldsymbol{y})$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda}\int_{\Pi}D_{[\lambda
x_{1}]}(u)\overline{D_{[\lambda
y_{1}]}(u)}\mathrm{d}u\times\int_{\Pi}f(0,v)D_{[\lambda^{\gamma}x_{2}]}(v)\overline{D_{[\lambda^{\gamma}y_{2}]}(v)}\mathrm{d}v=:J_{1}\times
J_{2},$

where

$\displaystyle J_{1}$ $\displaystyle\to$
$\displaystyle\int_{\mathbb{R}}\frac{(1-\mathrm{e}^{\mathrm{i}x_{1}u})(1-\mathrm{e}^{-\mathrm{i}y_{1}u})}{|u|^{2}}\mathrm{d}u=(x_{1}\wedge
y_{1})\int_{\mathbb{R}}\frac{|1-\mathrm{e}^{\mathrm{i}u}|^{2}}{|u|^{2}}\mathrm{d}u$

follows by the DCT, and

$\displaystyle J_{2}$ $\displaystyle=$
$\displaystyle\int_{\Pi}\frac{f(0,v)}{|1-\mathrm{e}^{\mathrm{i}v}|^{2}}\big{(}1-\mathrm{e}^{\mathrm{i}[\lambda^{\gamma}x_{2}]v}-\mathrm{e}^{-\mathrm{i}[\lambda^{\gamma}y_{2}]v}+\mathrm{e}^{\mathrm{i}([\lambda^{\gamma}x_{2}]-[\lambda^{\gamma}y_{2}])v}\big{)}\mathrm{d}v$
$\displaystyle\to$
$\displaystyle\int_{\Pi}\frac{f(0,v)}{|1-\mathrm{e}^{\mathrm{i}v}|^{2}}\mathrm{d}v\times\begin{cases}2,&x_{2}=y_{2},\\\
1,&x_{2}\neq y_{2},\end{cases}$

by the Lebesgue-Riemann theorem [26, Thm.1.2] and the integrability of
$f(0,v)|1-\mathrm{e}^{\mathrm{i}v}|^{-2}$. Whence, the asymptotic constant in
(4.13) equals

$\kappa_{-}^{2}\ :=\
2\int_{\mathbb{R}}|(1-\mathrm{e}^{\mathrm{i}u})/u|^{2}\mathrm{d}u\times\int_{\Pi}f(0,v)|1-\mathrm{e}^{\mathrm{i}v}|^{-2}\mathrm{d}v.$
(4.14)

To show the second relation in (4.13), similarly as in the subcase
$\gamma>\upsilon_{1}$ above write $|\tilde{f}_{2}(\boldsymbol{u})|\leq
C\sum_{k=1}^{3}|h_{2,k}(\boldsymbol{u})|$ and
$\tilde{R}_{2}(\boldsymbol{x},\boldsymbol{x})\leq
C\sum_{k=1}^{3}\tilde{R}_{3,k}(\boldsymbol{x},\boldsymbol{x})$ accordingly,
where
$h_{2,1}(\boldsymbol{u}):=\rho(\boldsymbol{u})-\rho(0,u_{2})=|u_{1}|^{\upsilon_{1}},$
$h_{2,2}(\boldsymbol{u}):=$ $\rho(0,u_{2})(L(\boldsymbol{u})-L(0,u_{2}))$ and
$h_{2,3}(\boldsymbol{u}):=\rho(\boldsymbol{u})\delta(\boldsymbol{u})$, where
$\delta(\boldsymbol{u})$ is a bounded function tending to $0$ as
$|\boldsymbol{u}|\to 0$. Then
$\tilde{R}_{2,3}(\boldsymbol{x},\boldsymbol{x})=o(1)$,

$\displaystyle|\tilde{R}_{2,1}(\boldsymbol{x},\boldsymbol{x})|$
$\displaystyle\leq$ $\displaystyle
C\lambda^{-1}\int_{\Pi^{2}}|u_{1}|^{\upsilon_{1}}\frac{|1-\mathrm{e}^{\mathrm{i}[\lambda
x_{1}]u_{1}}|^{2}|1-\mathrm{e}^{\mathrm{i}[\lambda^{\gamma}x_{2}]u_{2}}|^{2}}{|1-\mathrm{e}^{\mathrm{i}u_{1}}|^{2}|1-\mathrm{e}^{\mathrm{i}u_{2}}|^{2}}\mathrm{d}\boldsymbol{u}$
(4.15) $\displaystyle\leq$ $\displaystyle
C\lambda^{\gamma-\upsilon_{1}}\int_{\mathbb{R}}|u_{1}|^{\upsilon_{1}-2}|1-\mathrm{e}^{\mathrm{i}x_{1}u_{1}}|^{2}\mathrm{d}u_{1}\times\int_{\mathbb{R}}|u_{2}|^{-2}|1-\mathrm{e}^{\mathrm{i}x_{2}u_{2}}|^{2}\mathrm{d}u_{2}$
$\displaystyle=$ $\displaystyle O(\lambda^{\gamma-\upsilon_{1}})=o(1)$

and

$\displaystyle|\tilde{R}_{2,2}(\boldsymbol{x},\boldsymbol{x})|$
$\displaystyle\leq$ $\displaystyle
C\int_{\Pi}|u_{2}|^{\upsilon_{2}-2}\mathrm{d}u_{2}\int_{\mathbb{R}}\big{|}L\big{(}\frac{u_{1}}{\lambda},u_{2})-L(0,u_{2})\big{|}|(1-\mathrm{e}^{\mathrm{i}x_{1}u_{1}})/u_{1}|^{2}\mathrm{d}u_{1},=o(1),$
(4.16)

proving (4.13).

Subcase $\gamma=\upsilon_{1}$. We prove (2.3) with
$V_{0}=\kappa_{+}B_{H^{+}_{1},1/2}+\kappa_{-}B_{1/2,0}$ a sum of independent
limits in subcases $\gamma>\upsilon_{1}$ and $\gamma<\upsilon_{1}$. Split
$f(\boldsymbol{u})=f_{1}(\boldsymbol{u})+f_{2}(\boldsymbol{u})+\tilde{f}_{12}(\boldsymbol{u}),\tilde{f}_{12}(\boldsymbol{u}):=f(\boldsymbol{u})-f(u_{1},0)-f(0,u_{2})$
and, accordingly,
$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})=R_{1}(\boldsymbol{x},\boldsymbol{y})+R_{2}(\boldsymbol{x},\boldsymbol{y})+\tilde{R}_{12}(\boldsymbol{x},\boldsymbol{y})$.
Note $H^{+}_{1}+(\upsilon_{1}/2)=1/2$. The convergences
$R_{1}(\boldsymbol{x},\boldsymbol{y})\to\kappa^{2}_{+}\mathrm{E}B_{H^{+}_{1},1/2}(\boldsymbol{x},\boldsymbol{y})B_{H^{+}_{1},1/2}(\boldsymbol{x},\boldsymbol{y})$
and
$R_{2}(\boldsymbol{x},\boldsymbol{y})\to\kappa^{2}_{-}\mathrm{E}B_{1/2,0}(\boldsymbol{x},\boldsymbol{y})B_{1/2,0}(\boldsymbol{x},\boldsymbol{y})$
were proved in subcases $\gamma>\gamma_{0},\upsilon_{i}<1,i=1,2$ and
$\gamma<\gamma_{0},\upsilon_{1}<1<\upsilon_{2}$, respectively. It remains to
show

$\displaystyle\tilde{R}_{12}(\boldsymbol{x},\boldsymbol{x})\to 0.$ (4.17)

Write $f(\boldsymbol{u})=f_{0}(\boldsymbol{u})T(\boldsymbol{u})$, where
$f_{0}(\boldsymbol{u})=\rho(\boldsymbol{u})L(\boldsymbol{u})$ and
$T(\boldsymbol{u}):=\frac{f(\boldsymbol{u})}{f_{0}(\boldsymbol{u})},\boldsymbol{u}\in\Pi^{2}$
are a continuous functions, see Assumption (F)ND. Then,
$\tilde{f}_{12}(\boldsymbol{u})=\sum_{i=1}^{2}\tilde{f}_{12,i}(\boldsymbol{u})$,
where

$\displaystyle\tilde{f}_{12,1}(\boldsymbol{u}):=|u_{1}|^{\upsilon_{1}}(\widetilde{T}(\boldsymbol{u})-\widetilde{T}(u_{1},0)),\qquad\tilde{f}_{12,2}(\boldsymbol{u}):=|u_{2}|^{\upsilon_{2}}(\widetilde{T}(\boldsymbol{u})-\widetilde{T}(0,u_{2})),\quad\widetilde{T}(\boldsymbol{u}):=L(\boldsymbol{u})T(\boldsymbol{u}).$

Accordingly,
$\tilde{R}_{12}(\boldsymbol{x},\boldsymbol{x})=\sum_{i=1}^{2}\tilde{R}_{12,i}(\boldsymbol{x},\boldsymbol{x})$.
Then

$\displaystyle\tilde{R}_{12,1}(\boldsymbol{x},\boldsymbol{x})$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{2}}|u_{1}|^{\upsilon_{1}}\delta_{\lambda,1}(\boldsymbol{u})\prod_{j=1}^{2}|(1-\mathrm{e}^{\mathrm{i}u_{j}x_{j}})/u_{j}|^{2}\mathrm{d}\boldsymbol{u},$
(4.18)

where
$\delta_{\lambda,1}(\boldsymbol{u}):=\big{|}\widetilde{T}\big{(}\frac{u_{1}}{\lambda},\frac{u_{2}}{\lambda^{\upsilon_{1}}}\big{)}-\widetilde{T}\big{(}\frac{u_{1}}{\lambda},0\big{)}\big{|}\mathbb{I}(\boldsymbol{u}\in\lambda\Pi\times\lambda^{\upsilon_{1}}\Pi)$
is bounded uniformly in $\lambda>1$ and the integral on the r.h.s. with
$\delta_{\lambda,1}(\boldsymbol{u})$ replaced by 1 converges due to
$\upsilon_{1}<1$. Also note that $\delta_{\lambda,1}(\boldsymbol{u})\to 0$ as
$T(\boldsymbol{u}),\boldsymbol{u}\in\Pi^{2}$ is continuous and
$L\big{(}\frac{u_{1}}{\lambda},\frac{u_{2}}{\lambda^{\upsilon_{1}}}\big{)}-L\big{(}\frac{u_{1}}{\lambda},0\big{)}\to
0$ due to $\upsilon_{2}>1$. This proves (4.17) for
$\tilde{R}_{12,1}(\boldsymbol{x},\boldsymbol{x})$ instead of
$\tilde{R}_{12}(\boldsymbol{x},\boldsymbol{x})$. The proof of
$\tilde{R}_{12,2}(\boldsymbol{x},\boldsymbol{x})\to 0$ is similar to (4.16),
with $\big{|}L\big{(}\frac{u_{1}}{\lambda},u_{2})-L(0,u_{2})\big{|}$ there
replaced by
$\delta_{\lambda,2}(\boldsymbol{u}):=\big{|}\widetilde{T}\big{(}\frac{u_{1}}{\lambda},u_{2})-\widetilde{T}(0,u_{2})\big{|}\to
0$ in view of the above mentioned properties of $L(\boldsymbol{u})$ and
$T(\boldsymbol{u})$.

Case $\upsilon_{i}>1,i=1,2,\gamma_{0}=1$.  By symmetry, it suffices to
consider the case $\gamma\leq 1$. Split
$f(\boldsymbol{u})=f_{1}(\boldsymbol{u})+f_{2}(\boldsymbol{u})+\tilde{f}_{12}(\boldsymbol{u}),$
$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})=R_{1}(\boldsymbol{x},\boldsymbol{y})+R_{2}(\boldsymbol{x},\boldsymbol{y})+\tilde{R}_{12}(\boldsymbol{x},\boldsymbol{y})$
as in the previous case. Then
$R_{2}(\boldsymbol{x},\boldsymbol{y})\to\kappa^{2}_{-}\mathrm{E}B_{1/2,0}(\boldsymbol{x})B_{1/2,0}(\boldsymbol{y})$
as in (4.13) while $R_{1}(\boldsymbol{x},\boldsymbol{x})\leq
C\lambda^{\gamma-1}\int_{\Pi}|u_{1}|^{\upsilon_{1}-2}\mathrm{d}u_{1}$
$\times\int_{\mathbb{R}}|(1-\mathrm{e}^{\mathrm{i}u_{2}x_{2}})/u_{2}|^{2}\mathrm{d}u_{2}=O(\lambda^{\gamma-1})=o(1)$
is negligible when $\gamma<1$; for $\gamma=1$ we have
$R_{1}(\boldsymbol{x},\boldsymbol{y})\to\kappa^{2}_{+}\mathrm{E}B_{0,1/2}(\boldsymbol{x})B_{0,1/2}(\boldsymbol{y})$
analogously to (4.13) with

$\kappa_{+}^{2}\ :=\
2\int_{\mathbb{R}}|(1-\mathrm{e}^{\mathrm{i}u})/u|^{2}\mathrm{d}u\times\int_{\Pi}f(v,0)|1-\mathrm{e}^{\mathrm{i}v}|^{-2}\mathrm{d}v.$
(4.19)

The proof of $\tilde{R}_{12}(\boldsymbol{x},\boldsymbol{x})\to 0$ for
$\gamma\leq 1$ follows as in (4.16). This ends the proof of Part I.

Part II. Case $\upsilon_{1}=1,\upsilon_{2}<1,\gamma_{0}=1/\upsilon_{2}$.  Let
first
$\gamma>1/\upsilon_{2},d^{2}_{\lambda,\gamma}=\lambda^{\gamma}\log_{+}\lambda$.
Split $f(\boldsymbol{u})=f_{1}(\boldsymbol{u})+\tilde{f}_{1}(\boldsymbol{u})$
and
$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})=R_{1}(\boldsymbol{x},\boldsymbol{y})+\tilde{R}_{1}(\boldsymbol{x},\boldsymbol{y})$
as in the case $\upsilon_{1}<1<\upsilon_{2}$ above. Then

$\displaystyle R_{1}(\boldsymbol{x},\boldsymbol{y})$ $\displaystyle\sim$
$\displaystyle\frac{1}{\log\lambda}\int_{\lambda\Pi}\frac{(1-\mathrm{e}^{\mathrm{i}x_{1}u_{1}})(1-\mathrm{e}^{-\mathrm{i}y_{1}u_{1}})}{|u_{1}|^{2}}\lambda
f(u_{1}/\lambda,0)\mathrm{d}u_{1}$ (4.20)
$\displaystyle\times\int_{\lambda^{\gamma}\Pi}\frac{(1-\mathrm{e}^{\mathrm{i}x_{2}u_{2}})(1-\mathrm{e}^{-\mathrm{i}y_{2}u_{2}})}{|u_{2}|^{2}}\mathrm{d}u_{2}\
=:\ J_{1}\times J_{2},$

where $J_{2}\to\kappa^{2}(x_{2}\wedge
y_{2}),\kappa^{2}:=\int_{\mathbb{R}}|(1-\mathrm{e}^{\mathrm{i}u})/u|^{2}\mathrm{d}u$
and we need to show the limit of $J_{1}$. We have $\lambda
f(u_{1}/\lambda,0)=\lambda
L(1,0)|u_{1}/\lambda|(1+\delta(u_{1}/\lambda))=L(1,0)|u_{1}|(1+\delta(u_{1}/\lambda))$
where $\delta(u)$ is a bounded function tending to 0 as $u\to 0$. Therefore,
$J_{1}=J^{\prime}_{1}+J^{\prime\prime}_{1}$, where

$J^{\prime}_{1}:=\frac{2L(1,0)}{\log\lambda}{\rm
Re}\int_{0}^{\lambda\pi}(1-\mathrm{e}^{\mathrm{i}x_{1}u})(1-\mathrm{e}^{-\mathrm{i}y_{1}u})u^{-1}\mathrm{d}u\
\to\ 2L(1,0)\begin{cases}2,&x_{1}=y_{1},\\\ 1,&x_{1}\neq y_{1}\end{cases}$

as
$\lim_{\lambda\to\infty}\int_{1}^{\lambda}\mathrm{e}^{\mathrm{i}xu}u^{-1}\mathrm{d}u$
exists for any $x\neq 0$, and $|J^{\prime\prime}_{1}|\leq
C(\log\lambda)^{-1}`\int_{0}^{\lambda\pi}(u\wedge
1)^{2}u^{-1}\delta(u/\lambda)\mathrm{d}u\to 0$ follows as in (3.7) by
splitting the last integral over sets $u/\lambda<\epsilon$ and $\epsilon\leq
u/\lambda\leq\pi$. Whence,
$R_{1}(\boldsymbol{x},\boldsymbol{y})\to\kappa_{+}^{2}\mathrm{E}B_{0,1/2}(\boldsymbol{x})B_{0,1/2}(\boldsymbol{y})$,
where

$\kappa^{2}_{+}=4L(1,0)\kappa^{2}=4L(1,0)\int_{\mathbb{R}}|(1-\mathrm{e}^{\mathrm{i}u})/u|^{2}\mathrm{d}u.$
(4.21)

To show that $\tilde{R}_{1}(\boldsymbol{x},\boldsymbol{x})$ is negligible, use
the decomposition of $\tilde{f}_{1}(\boldsymbol{u})$ following (4.10) so that
$|\tilde{R}_{1}(\boldsymbol{x},\boldsymbol{x})|\leq
C\sum_{k=1}^{3}\tilde{R}_{1,k}(\boldsymbol{x},\boldsymbol{x})$ where
$\tilde{R}_{1,k}(\boldsymbol{x},\boldsymbol{x})=J_{\lambda,k}/\log\lambda$ and
$J_{\lambda,k}$ are as in the aforementioned proof. Then
$\tilde{R}_{1,1}(\boldsymbol{x},\boldsymbol{x})=o(1)$ as in (4.11) while
$\tilde{R}_{1,2}(\boldsymbol{x},\boldsymbol{x})\leq
C\int_{\mathbb{R}}(1+u_{2}^{2})^{-1}\delta_{\lambda}(u_{2})\mathrm{d}u_{2}$,
c.f. (4), with

$\delta_{\lambda}(u_{2}):=\frac{1}{\log\lambda}\int_{|u_{1}|\leq\lambda\pi}(1+|u_{1}|)^{-1}\big{|}L\big{(}\frac{u_{1}}{\lambda},\frac{u_{2}}{\lambda^{\gamma}}\big{)}-L\big{(}\frac{u_{1}}{\lambda},0\big{)}\big{|}\mathrm{d}u_{1}$

bounded and tending to 0 as $\lambda\to\infty$ for any fixed $u_{2}\neq 0$ by
continuity of $\tilde{L}$ due to $\gamma\upsilon_{2}>1$. Finally,
$\tilde{R}_{1,3}(\boldsymbol{x},\boldsymbol{x})\leq
C(\log\lambda)^{-1}\int_{\lambda\Pi}(1+|u_{1}|)^{-1}\mathrm{d}u_{1}\int_{\mathbb{R}}\delta(u_{1}/\lambda,u_{2}/\lambda^{\gamma})(1+u_{2}^{2})^{-1}\mathrm{d}u_{2}+o(1)=o(1)$
follows by splitting the last integral over $|u_{1}|<\epsilon\lambda$ and
$|u_{1}|>\epsilon\lambda$, using
$\int_{\epsilon\lambda}^{\pi\lambda}u_{1}^{-1}\mathrm{d}u_{1}=\log(1/\epsilon)<\infty$
for any small $\epsilon>0$.

Next, let
$\gamma<1/\upsilon_{2},d^{2}_{\lambda,\gamma}=\lambda^{2H(\gamma)},2H(\gamma)=1+\gamma(1-\upsilon_{2})$.
Note $\lambda^{\gamma\upsilon_{2}}f(u_{1}/\lambda,u_{2}/\lambda^{\gamma})\to
f_{0}(0,u_{2})=L(0,1)|u_{2}|^{\upsilon_{2}}$. Then
$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})\to\kappa^{2}_{-}\mathrm{E}B_{1/2,H^{-}_{2}}(\boldsymbol{x})B_{1/2,H^{-}_{2}}(\boldsymbol{y})$
follows as in the case $\upsilon_{1}\wedge\upsilon_{2}<1,\gamma<\gamma_{0}$,
with
$\kappa^{2}_{-}=L(0,1)\int_{\mathbb{R}^{2}}|u_{2}|^{\upsilon_{2}}\prod_{j=1}^{2}|(1-\mathrm{e}^{{\mathrm{i}}u_{j}})/u_{j}|^{2}\mathrm{d}\boldsymbol{u},$
c.f. (4.9).

Let $\gamma=1/\upsilon_{2},$ then
$d^{2}_{\lambda,\gamma}=\lambda^{2H(\gamma)}\log\lambda,\,2H(\gamma)=\frac{1}{\upsilon_{2}}$.
We have

$\displaystyle R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle\sim$
$\displaystyle\frac{1}{\log\lambda}\int_{\lambda\Pi\times\lambda^{1/\upsilon_{2}}\Pi}\prod_{i=1}^{2}\frac{(1-\mathrm{e}^{\mathrm{i}x_{i}u_{i}})(1-\mathrm{e}^{-\mathrm{i}y_{i}u_{i}})}{|u_{i}|^{2}}\lambda
f(u_{1}/\lambda,u_{2}/\lambda^{1/\upsilon_{2}})\mathrm{d}\boldsymbol{u}$
(4.22)

where $\lambda
f(u_{1}/\lambda,u_{2}/\lambda^{1/\upsilon_{2}})=f_{0}(\boldsymbol{u})(1+\delta(u_{1}/\lambda,u_{2}/\lambda^{1/\upsilon_{2}}))$
$=\sum_{k=1}^{3}f_{k}(\boldsymbol{u}),$
$f_{1}(\boldsymbol{u}):=L(\boldsymbol{u})|u_{1}|,f_{2}(\boldsymbol{u}):=L(\boldsymbol{u})|u_{2}|^{\upsilon_{2}},$
$f_{3}(\boldsymbol{u}):=f_{0}(\boldsymbol{u})\delta(u_{1}/\lambda,u_{2}/\lambda^{1/\upsilon_{2}}),$
and $\lim_{\boldsymbol{u}\to{\bf 0}}\delta(\boldsymbol{u})=0$. Accordingly,
$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})\sim\sum_{k=1}^{3}R_{k}(\boldsymbol{x},\boldsymbol{y})$;
we will show that $R_{1}(\boldsymbol{x},\boldsymbol{y})$ is the main term and
$R_{k}(\boldsymbol{x},\boldsymbol{y})\to 0,k=2,3$. Indeed,

$\displaystyle R_{1}(\boldsymbol{x},\boldsymbol{y})$ $\displaystyle=$
$\displaystyle\frac{1}{\log\lambda}\int_{-\lambda\pi}^{\lambda\pi}\frac{(1-\mathrm{e}^{\mathrm{i}x_{1}u})(1-\mathrm{e}^{-\mathrm{i}y_{1}u})}{|u|}h_{\lambda}(u)\mathrm{d}u,$

where
$h_{\lambda}(u):=\int_{|w|\leq\lambda^{1/\upsilon_{2}}\pi}(1-\mathrm{e}^{\mathrm{i}x_{2}w})(1-\mathrm{e}^{-\mathrm{i}y_{2}w})|w|^{-2}L(u,w)\mathrm{d}w\to
h(u)\,(\lambda\to\infty)$ and where

$\displaystyle h(u)$ $\displaystyle:=$
$\displaystyle\int_{\mathbb{R}}\frac{(1-\mathrm{e}^{\mathrm{i}x_{2}w})(1-\mathrm{e}^{-\mathrm{i}y_{2}w})}{|w|^{2}}L(u,w)\mathrm{d}w$
$\displaystyle\to$ $\displaystyle
L(1,0)\int_{\mathbb{R}}\frac{(1-\mathrm{e}^{\mathrm{i}x_{2}w})(1-\mathrm{e}^{-\mathrm{i}y_{2}w})}{|w|^{2}}\mathrm{d}w=L(1,0)\kappa^{2}(x_{2}\wedge
y_{2})$

as $|u|\to\infty$, see (2.16), (2.17) for the last equality. Whence, the
convergence

$R^{\prime}_{1}(\boldsymbol{x},\boldsymbol{y})\ :=\
\frac{1}{\log\lambda}\int_{-\lambda\pi}^{\lambda\pi}\frac{(1-\mathrm{e}^{\mathrm{i}x_{1}u})(1-\mathrm{e}^{-\mathrm{i}y_{1}u})}{|u|}h(u)\mathrm{d}u\
\to\
\kappa_{+}^{2}\mathrm{E}B_{0,1/2}(\boldsymbol{x})B_{0,1/2}(\boldsymbol{y})$

with $\kappa^{2}_{+}$ in (4.21) follows as in (4.20) and the same limit for
$R_{1}(\boldsymbol{x},\boldsymbol{y})$ requires few changes. Next,
$|R_{2}(\boldsymbol{x},\boldsymbol{x})|\leq
C(\log\lambda)^{-1}\int_{\mathbb{R}^{2}}(1+u_{1}^{2})^{-1}(1+u_{2}^{2})^{-1}|u_{2}|^{\upsilon_{2}}\mathrm{d}\boldsymbol{u}=O(1/\log\lambda)$.
Finally, $|R_{3}(\boldsymbol{x},\boldsymbol{x})|\leq
C(\log\lambda)^{-1}\big{(}\int_{0}^{\lambda\pi}(1+u)^{-1}\int_{\mathbb{R}}(1+w^{2})^{-1}\delta(u/\lambda,w/\lambda^{1/\upsilon_{2}})\mathrm{d}w+O(1)\big{)}\to
0$ follows as in the case $\gamma>1/\upsilon_{2}$.

Case $\upsilon_{1}=1,\upsilon_{2}>1,\gamma_{0}=1$.  Let $\gamma\geq
1,d^{2}_{\lambda,\gamma}=\lambda^{\gamma}\log_{+}\lambda$. Then

$\displaystyle R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle\sim$
$\displaystyle\frac{1}{\log\lambda}\int_{\lambda\Pi\times\lambda^{\gamma}\Pi}\prod_{i=1}^{2}\frac{(1-\mathrm{e}^{\mathrm{i}x_{i}u_{i}})(1-\mathrm{e}^{-\mathrm{i}y_{i}u_{i}})}{|u_{i}|^{2}}\lambda
f(u_{1}/\lambda,u_{2}/\lambda^{\gamma})\mathrm{d}\boldsymbol{u}$

where $\lambda f(u_{1}/\lambda,u_{2}/\lambda^{\gamma})\to
f_{0}(u_{1},0)=L(1,0)|u|$ due to $\gamma\upsilon_{2}>1$. Then
$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})\to\kappa^{2}_{+}\mathrm{E}B_{0,1/2}(\boldsymbol{x})B_{0,1/2}(\boldsymbol{y})$
as in (4.20) with $\kappa^{2}_{+}$ in (4.21). Next, let $\gamma<1$. Then
$d_{\lambda,\gamma}=\lambda^{1/2}$ and

$\displaystyle R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle\sim$
$\displaystyle\int_{-\pi\lambda}^{\pi\lambda}\frac{(1-\mathrm{e}^{\mathrm{i}x_{1}u})(1-\mathrm{e}^{-\mathrm{i}y_{1}u})}{|u|^{2}}\mathrm{d}u\times\int_{\Pi}\frac{(1-\mathrm{e}^{\mathrm{i}\lambda^{\gamma}x_{2}w})(1-\mathrm{e}^{-\mathrm{i}\lambda^{\gamma}y_{2}w})}{|1-\mathrm{e}^{\mathrm{i}v}|^{2}}f(u/\lambda,w)\mathrm{d}w$
$\displaystyle\sim$
$\displaystyle\kappa^{2}_{-}\,\mathrm{E}B_{1/2,0}(\boldsymbol{x})B_{1/2,0}(\boldsymbol{y})$

as in (4.13) with $\kappa^{2}_{-}$ in (4.14).

Case $\upsilon_{1}=\upsilon_{2}=\gamma_{0}=1$.  The convergence in (2.3) for
$\gamma\neq 1$ leading to limits $V_{+}=\kappa_{+}B_{0,1/2}$ and
$V_{-}=\kappa_{-}B_{1/2,0}$ by symmetry follow as in the case
$\upsilon_{1}=1,\upsilon_{2}>1$, with with $\kappa^{2}_{+}$ in (4.21) and
$\kappa^{2}_{-}=4L(0,1)\kappa^{2}$. Let us prove that for $\gamma=1$ (2.3)
tends to the sum of the latter limits leading to

$\displaystyle V_{0}$ $\displaystyle=$
$\displaystyle\kappa_{+}B_{0,1/2}+\kappa_{-}B_{1/2,0},\quad\text{with}\quad
d^{2}_{\lambda,\gamma}=\lambda\log_{+}\lambda,$

where $B_{0,1/2}$ and $B_{1/2,0}$ are mutually independent. Proceeding as in
(4.22) we see that $f_{\lambda}(\boldsymbol{u}):=\lambda
f(u_{1}/\lambda,u_{2}/\lambda)$ $\to
f_{0}(\boldsymbol{u})=L(\boldsymbol{u})(|u_{1}|+|u_{2}|)$ and
$R_{\lambda,\gamma}(\boldsymbol{x},\boldsymbol{y})$ behaves asymptotically as
the sum of two terms

$\displaystyle R_{k}(\boldsymbol{x},\boldsymbol{y})$ $\displaystyle:=$
$\displaystyle\frac{1}{\log\lambda}\int_{(\lambda\Pi)^{2}}L(u_{1},u_{2})|u_{k}|\prod_{j=1}^{2}\frac{(1-\mathrm{e}^{\mathrm{i}x_{j}u_{j}})(1-\mathrm{e}^{-\mathrm{i}y_{j}u_{j}})}{|u_{j}|^{2}}\mathrm{d}\boldsymbol{u},\quad
k=1,2$

which tend to
$\kappa^{2}_{+}\mathrm{E}B_{0,1/2}(\boldsymbol{x})B_{0,1/2}(\boldsymbol{y})$
and
$\kappa^{2}_{-}\mathrm{E}B_{1/2,0}(\boldsymbol{x})B_{1/2,0}(\boldsymbol{y})$,
respectively. We also need to check that the term
$R_{3}(\boldsymbol{x},\boldsymbol{y})$ corresponding to
$f_{\lambda}(\boldsymbol{u})-f_{0}(\boldsymbol{u})$ is negligible, viz.,
$|R_{3}(\boldsymbol{x},\boldsymbol{y})|\leq C(\log\lambda)^{-1}$
$\int_{(\lambda\Pi)^{2}}(|u_{1}|+|u_{2}|)$
$\delta(u_{1}/\lambda,u_{2}/\lambda)$
$\prod_{j=1}^{2}(1+|u_{j}|^{2})^{-1}\mathrm{d}\boldsymbol{u}\to 0$. We omit
these details being similar to (4.22). This ends the proof of Part II and
Theorem 2, too. $\Box$

######  Remark 2.

Following the terminology in time series [13], the asymptotic constants
$\kappa^{2}_{\pm}$ may be dubbed ‘long-range variances’. It is notable that
the only case when $\kappa^{2}_{\pm}$ depend on the spectral density outside
of the origin are (4.14), (4.19) corresponding to $H^{\pm}_{i}=0$ or spectrum
ND under ‘edge effects’, see Remark 3. Obviously, these expressions for
$\kappa^{2}_{\pm}$ make sense for continuous $f$ (the continuity can be
relaxed) but not for an arbitrary integrable or bounded $f$. On the other
hand, continuity of $f$ is a consequence of summability of covariance function
hence occurs under covariance SRD and ND.

##  5 Long-range negative and hyperbolic dependence

Recall the asymptotic form of the spectral density under Assumption (F)LRND,2:

$f(\boldsymbol{u})\ \sim\
f_{0}(\boldsymbol{u})=\frac{L(\boldsymbol{u})}{|u_{1}|^{\upsilon_{1}}+|u_{2}|^{\upsilon_{2}}}\qquad|\boldsymbol{u}|\to
0,$ (5.1)

where
$L(\boldsymbol{u})=L(-\boldsymbol{u}),\boldsymbol{u}\in\mathbb{R}^{2}_{+}$ is
a continuous generalized invariant function such that

$L(\boldsymbol{u})\sim\ell\Big{(}\frac{|u_{2}|}{\rho(\boldsymbol{u})^{1/\upsilon_{2}}}\Big{)}^{\mu}\qquad(u_{2}\to
0)$ (5.2)

for some $\mu\in(0,1),\ell>0$. Define

$\displaystyle H^{+}_{1}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\Big{(}1+\big{(}(\upsilon_{1}+\frac{\mu\upsilon_{1}}{\upsilon_{2}})\wedge
1\big{)}\Big{)},\quad
H^{+}_{2}:=\frac{1}{2}\Big{(}1-\big{(}\mu\wedge(\frac{\upsilon_{2}}{\upsilon_{1}}-\upsilon_{2})\big{)}\Big{)},$
(5.3) $\displaystyle H^{-}_{1}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\big{(}1+\upsilon_{1}-\frac{\upsilon_{1}}{\upsilon_{2}\vee
1}\big{)},\hskip 56.9055ptH^{-}_{2}:=\frac{1}{2}\big{(}1+(\upsilon_{2}\wedge
1)\big{)}$

Note $H^{-}_{i},i=1,2$ in (5.3) are the same as in Theorem 1, (3.1), whereas
the expressions for $H^{+}_{i},i=1,2$ in (5.3) and (3.1) agree if and only if
$\mu=0$. Also note that $H^{\pm}_{1},H^{-}_{2}\in[1/2,1]$ while
$H^{+}_{2}=\frac{1}{2}(1+\upsilon_{2}-\frac{\upsilon_{2}}{\upsilon_{1}})\in[1/2,1]$
for $\upsilon_{1}\geq 1$; for $\upsilon_{1}\leq 1$ we have
$H^{+}_{2}\in(0,1/2]$. Finally, $H(\gamma)$ in (5.5) is a continuous function
of $\gamma,\upsilon_{i},i=1,2,\mu,$ its value
$H(\gamma_{0})=\frac{1}{2}(1+\upsilon_{1}+\frac{\upsilon_{1}}{\upsilon_{2}})$
at $\gamma=\gamma_{0}:=\frac{\upsilon_{1}}{\upsilon_{2}}$ being the same as in
Theorem 1 independently of $\mu$.

The following theorem excludes some particular cases of parameters
$\mu,\upsilon_{i},i=1,2$ which may require extra logarithmic normalizing
factor. It also leaves open the question about the scaling limits when both
Assumptions (F)${}_{\rm LRN\\!D,1}$ and (F)${}_{\rm LRN\\!D,2}$ are satisfied.

######  Theorem 3.

Let $X$ in (1.1) be a stationary linear RF on $\mathbb{Z}^{2}$ in (1.3) with
spectral density $f$ satisfying Assumption (F)${}_{LRN\\!D,2}$. In addition,
let $\upsilon_{1}\neq 1,\upsilon_{1}+\frac{\upsilon_{2}}{\upsilon_{1}}\neq 1$.
Then:

  * •

The scaling limits in (1.2) exist for any $\gamma>0$ and satisfy (2.1) with
$\gamma_{0}=\frac{\upsilon_{1}}{\upsilon_{2}}$ and the unbalanced limits are
given by

$\displaystyle V_{+}$ $\displaystyle:=$
$\displaystyle\kappa_{+}B_{H^{+}_{1},H^{+}_{2}},\qquad V_{-}\ :=\
\kappa_{-}B_{H^{-}_{1},H^{-}_{2}},$ (5.4)

where $H^{\pm}_{i},i=1,2$ as in (5.3). The asymptotic constant $\kappa_{+}$ is
defined in (5.6), (5.7), whereas $\kappa_{-}$ is the same as in Theorem 1.

  * •

The well-balanced limit $V_{0}:=V_{0,1/\rho}$ is given in (2.12).

  * •

The normalization in (1.2) is given by
$d_{\lambda,\gamma}:=\lambda^{H(\gamma)}$ with

$\displaystyle H(\gamma)$ $\displaystyle:=$
$\displaystyle\begin{cases}H^{+}_{1}+\gamma
H^{+}_{2},&\gamma\geq\gamma_{0},\\\ H^{-}_{1}+\gamma
H^{-}_{2},&\gamma\leq\gamma_{0}.\end{cases}$ (5.5)

  * •

The RF $X$ exhibits scaling transition at
$\gamma_{0}=\frac{\upsilon_{1}}{\upsilon_{2}}$.

Proof. Since (5.5) satisfy $H(\gamma)>1/2\,(\forall\,\gamma>0)$, the Lindeberg
criterion in (2.4) is satisfied as in Theorem 1. For $\gamma\leq\gamma_{0}$
the results of Theorem 3 and their proof completely agree with those of
Theorem 1. The subsequent proof on (2.3) is limited to $\gamma>\gamma_{0}$ and
split into three cases as follows.

Case $\upsilon_{1}+\frac{\upsilon_{1}}{\upsilon_{2}}<1$. According to
(5.1)-(5.2) for $\gamma>\upsilon_{1}/\upsilon_{2}$ we have that

$\displaystyle f(u_{1}/\lambda,u_{2}/\lambda^{\gamma})$ $\displaystyle\sim$
$\displaystyle\frac{\lambda^{\upsilon_{1}}}{|u_{1}|^{\upsilon_{1}}}\times\ell\Big{|}\frac{\frac{u_{2}}{\lambda^{\gamma}}}{|\frac{u_{1}}{\lambda}|^{\upsilon_{1}/\upsilon_{2}}}\Big{|}^{\mu}\
=\ \ell\lambda^{2H(\gamma)-1-\gamma}\prod_{j=1}^{2}|u_{j}|^{1-2H^{+}_{j}}$

point-wise in $\boldsymbol{u}=(u_{1},u_{2})\in\mathbb{R}^{2}_{0}$. Then, the
limit in (2.3) with $V_{\gamma}=V_{+}=\kappa_{+}B_{H^{+}_{1},H^{+}_{2}}$
follows similarly as in Theorem 1 with

$\displaystyle\kappa^{2}_{+}$ $\displaystyle=$
$\displaystyle\ell\prod_{j=1}^{2}\int_{\mathbb{R}}|(1-\mathrm{e}^{\mathrm{i}u})/u|^{2}|u|^{1-2H^{+}_{j}}\mathrm{d}u.$
(5.6)

Case $\upsilon_{1}<1<\upsilon_{1}+\frac{\upsilon_{1}}{\upsilon_{2}}$. Then
$(2H^{+}_{1},2H^{+}_{2})=(1+\upsilon_{1}+\frac{\mu\upsilon_{1}}{\upsilon_{2}},1-\mu)$
if $\mu<\frac{\upsilon_{2}}{\upsilon_{1}}-\upsilon_{2}$,
$=(2,1+\upsilon_{2}-\frac{\upsilon_{2}}{\upsilon_{1}})$ if
$\mu>\frac{\upsilon_{2}}{\upsilon_{1}}-\upsilon_{2}$. The proof of the limit
$V_{+}=\kappa_{+}B_{H^{+}_{1},H^{+}_{2}}$ for
$\mu<\frac{\upsilon_{2}}{\upsilon_{1}}-\upsilon_{2}$ is the same as in the
case $\upsilon_{1}+\frac{\upsilon_{1}}{\upsilon_{2}}<1$ above, and is omitted.
Next, let $\mu>\frac{\upsilon_{2}}{\upsilon_{1}}-\upsilon_{2}.$ Due to
$\gamma>\upsilon_{1}/\upsilon_{2}$ we see that

$\lambda^{-\gamma\upsilon_{2}}f\big{(}\frac{u_{1}}{\lambda^{\gamma\upsilon_{2}/\upsilon_{1}}},\frac{u_{2}}{\lambda^{\gamma}}\big{)}\to
f_{0}(\boldsymbol{u}),\quad\lambda^{-1}D_{[\lambda
x]}\big{(}\frac{u_{1}}{\lambda^{\gamma\upsilon_{2}/\upsilon_{1}}}\big{)}\to
x,\quad\lambda^{-\gamma}D_{[\lambda^{\gamma}x]}\big{(}\frac{u_{2}}{\lambda^{\gamma}}\big{)}\to\frac{1-\mathrm{e}^{\mathrm{i}xu_{2}}}{-\mathrm{i}u_{2}}$

point-wise for any $u_{i}\neq 0,i=1,2,x>0$. We also have
$\int_{\mathbb{R}}g_{+}(u_{1},u_{2})\mathrm{d}u_{1}=|u_{2}|^{1-2H^{+}_{2}}\int_{\mathbb{R}}f_{0}(u,1)\mathrm{d}u$
where the last integral converges due to (5.2) and
$\mu>\frac{\upsilon_{2}}{\upsilon_{1}}-\upsilon_{2}$. Then,
$V_{+}=\kappa_{+}B_{1,H^{+}_{2}}$ follows as in [24, Thm.3.1]. The asymptotic
constant $\kappa_{+}$ in both cases of $\mu$ is given by

$\displaystyle\kappa^{2}_{+}$ $\displaystyle=$
$\displaystyle\begin{cases}\ell\prod_{j=1}^{2}\int_{\mathbb{R}}|(1-\mathrm{e}^{\mathrm{i}u})/u|^{2}|u|^{1-2H^{+}_{j}}\mathrm{d}u,&\mu<\frac{\upsilon_{2}}{\upsilon_{1}}-\upsilon_{2},\\\
\int_{\mathbb{R}}\big{|}(1-\mathrm{e}^{\mathrm{i}v})/v\big{|}^{2}|v|^{1-2H^{+}_{2}}\mathrm{d}v\times\int_{\mathbb{R}}g_{+}(u,1)\mathrm{d}u,&\mu>\frac{\upsilon_{2}}{\upsilon_{1}}-\upsilon_{2}.\end{cases}$
(5.7)

Case $\upsilon_{1}>1$. In this case, the results do not depend on $\mu$ and
completely agree with those in Theorem 1, including the proof. This ends the
proof of Theorem 3. $\Box$

Next, we formalize the meaning of hyperbolic dependence mentioned in the
Introduction. Let

$f(\boldsymbol{u})=f_{0,{\rm
hyp}}(\boldsymbol{u})(1+o(1)),\quad|\boldsymbol{u}|\to
0,\quad\text{where}\quad f_{0,{\rm
hyp}}(\boldsymbol{u}):=L(\boldsymbol{u})\prod_{i=1}^{2}|u_{i}|^{-\upsilon_{i}},\qquad\boldsymbol{u}\in\Pi^{2},$
(5.8)

$|\upsilon_{i}|<1,i=1,2$ and

$\displaystyle
L(\boldsymbol{u}):=\tilde{L}\Big{(}\frac{u_{1}}{(|u_{1}|^{|\upsilon_{1}|}+|u_{2}|^{|\upsilon_{2}|})^{1/|\upsilon_{1}|}},\frac{u_{2}}{(|u_{1}|^{|\upsilon_{1}|}+|u_{2}|^{|\upsilon_{2}|})^{1/|\upsilon_{2}|}}\Big{)}$
(5.9)

is a generalized invariant function corresponding to generalized homogeneous
function
$\rho(\boldsymbol{u})=|u_{1}|^{|\upsilon_{1}|}+|u_{2}|^{|\upsilon_{2}|},\boldsymbol{u}=(u_{1},u_{2})\in\mathbb{R}^{2}_{0}$.
Let

$H_{i}:=\frac{1}{2}(1+\upsilon_{i}),\qquad i=1,2.$ (5.10)

The class in (5.8) includes (separately) fractionally integrated spectral
densities $\prod_{i=1}^{2}|1-\mathrm{e}^{\mathrm{i}u_{i}}|^{-\upsilon_{i}}$
which play an important role in spatial statistics [5, 14, 17]. If
$L(\boldsymbol{u})$ is separated from 0 and $\infty$, the spectral density in
(5.8) satisfies (spectrum) LRD, ND or LRND properties depending on the sign of
$\upsilon_{i},i=1,2$ except that it explodes/vanishes on the coordinate axis
$u_{i}=0$ when $\upsilon_{i}\neq 0$ as well and represents a different class
from those discussed in Theorems 1-3 Introduce a Gaussian RF

$\displaystyle V_{\rm 0,hyp}(\boldsymbol{x})$ $\displaystyle:=$
$\displaystyle\int_{\mathbb{R}^{2}}\prod_{j=1}^{2}\frac{1-\mathrm{e}^{{\mathrm{i}}u_{j}x_{j}}}{\mathrm{i}u_{j}}\,\sqrt{f_{\rm
0,hyp}(\boldsymbol{u})}Z(\mathrm{d}\boldsymbol{u}),\quad\boldsymbol{x}\in\mathbb{R}^{2}_{+},$
(5.11)

where $Z(\mathrm{d}\boldsymbol{u})$ is the same white noise as in (2.12). In
the case when $L(\boldsymbol{u})=\ell>0$ is a constant function, the RF
$V_{\rm 0,hyp}$ is a multiple of FBS: $V_{\rm 0,hyp}=\kappa\ell
B_{H_{1},H_{2}}$ with $H_{i},i=1,2$ given in (5.10) and $\kappa>0$ as in
(2.17).

######  Theorem 4.

be a stationary linear RF on $\mathbb{Z}^{2}$ in (1.3) with spectral density
$f$ in (5.8)-(5.9), where $\upsilon_{i}\in(-1,1),i=1,2$ and
$\tilde{L}(\boldsymbol{x})$ is a strictly positive continuous function. Then:

  * •

The scaling limits in (1.2) exist for any $\gamma>0$ and satisfy

$V_{\gamma}=\begin{cases}\kappa_{\gamma}B_{H_{1},H_{2}},&\gamma\neq\frac{|\upsilon_{1}|}{|\upsilon_{2}|},\\\
V_{\rm 0,hyp},&\gamma=\frac{|\upsilon_{1}|}{|\upsilon_{2}|},\end{cases}$
(5.12)

where $H_{i},i=1,2$ as in (5.10) and

$\kappa^{2}_{\gamma}:=\prod_{j=1}^{2}\int_{\mathbb{R}}|(1-\mathrm{e}^{\mathrm{i}u})/u|^{2}|u|^{1-2H_{j}}\mathrm{d}u\times\begin{cases}L(1,0),&\gamma>\frac{|\upsilon_{1}|}{|\upsilon_{2}|},\\\
L(0,1),&\gamma<\frac{|\upsilon_{1}|}{|\upsilon_{2}|}.\end{cases}$ (5.13)

  * •

The normalization in (1.2) is given by
$d_{\lambda,\gamma}:=\lambda^{H(\gamma)},H(\gamma):=H_{1}+\gamma H_{2}.$

  * •

The RF $X$ does not exhibit scaling transition.

We omit the proof of Theorem 4 since it resembles [24] and the previous
proofs. The gaussianity criterion (2.4) holds for ${\rm
sign}(\upsilon_{1})={\rm sign}(\upsilon_{2})$ as in Theorems 1 and 2; for
$\upsilon_{1}\upsilon_{2}<0$ from (5.8) we get
$|\widehat{a}(\boldsymbol{u})|\leq C\prod_{i=1}^{2}|u_{i}|^{-\upsilon_{i}/2}$
and (2.4) follows similarly. The absence of scaling transition at
$\gamma=\frac{|\upsilon_{1}|}{|\upsilon_{2}|}$ is clear from (5.12)-(5.13) and
the fact that $L(1,0)>0,L(0,1)>0$ according to the positivity assumption on
$L$ (the last conclusion may fail if $L(1,0)$ and/or $L(0,1)$ vanish).

We end the paper with two remarks on possible extensions of this work.

######  Remark 3.

More general scaling schemes and ‘edge effects’. In a more abstract setting, a
RF is often indexed by test functions, through which scaling operations are
defined; see [8, 9, 11]. Accordingly, one can study anisotropic scaling limits
of integrals

$S_{\lambda,\gamma}(\phi):=\int_{\mathbb{R}^{2}}\phi(\lambda^{-\Gamma}\boldsymbol{t})X(\lceil\boldsymbol{t}\rceil)\mathrm{d}\boldsymbol{t}$
(5.14)

involving $X$ in (1.3) extended to $\mathbb{R}^{2}$;
$\lceil\boldsymbol{t}\rceil:=(\lceil t_{1}\rceil,\lceil
t_{2}\rceil),\boldsymbol{t}=(t_{1},t_{2})\in\mathbb{R}^{2}$, and a re-scaled
function $\phi:\mathbb{R}^{2}\to\mathbb{R}$,
$\lambda^{-\Gamma}\boldsymbol{t}:=(\lambda^{-1}t_{1},\lambda^{-\gamma}t_{2}),\Gamma:={\rm
diag}(1,\gamma)$, for a given $\gamma>0$ and each $\phi$ from a class $\Phi$
of (test) functions. The sum in (1.1) corresponds to (5.14) with indicator
function
$\phi(\boldsymbol{t})=\mathbb{I}(\boldsymbol{t}\in]\boldsymbol{0},\boldsymbol{x}])$.
For (5.14), the scaling limit
$d^{-1}_{\lambda,\gamma}(S_{\lambda,\gamma}(\phi)-\mathrm{E}S_{\lambda,\gamma}(\phi))\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\ \end{array}V_{\gamma}(\phi)$ is a RF indexed by
$\phi\in\Phi$. Isotropic $(\gamma=1)$ scaling limits in spirit of (5.14) for
different classes of RF on $\mathbb{R}^{d}$ were studied in [9, 3, 15] and
other papers. Extending Theorems 1-4 to scaling limits of integral functionals
in (5.14) for suitable class $\Phi$ of test functions which include indicator
functions $\phi(\boldsymbol{t})=\mathbb{I}(\boldsymbol{t}\in A)$ of general
bounded sets $A\subset\mathbb{R}^{2}$ is an interesting problem. Of particular
interest is the case of ND RF $X$, where ‘edge effects’ can be expected
following [16, 28], leading to scaling limits $V_{\gamma}$ ‘living’ on the
boundary of $A$ or the discontinuity set of $\phi$.

######  Remark 4.

Incongruous scaling and dependence axis. For linear LRD RF $X$ in (1.3) [20]
define the dependence axis of $X$ as the direction on the plane along which
$a(\boldsymbol{t})$ decay at the smallest rate. In spectral terms, we may
define the dependence axis as a direction along which the spectral density
$f(\boldsymbol{x})$ grows at the fastest rate when $|\boldsymbol{x}|\to 0$.
Under Assumption (F)LRD with $\upsilon_{1}\neq\upsilon_{2}$ such dependence
axis coincides with one of the coordinate axes. The scaling in (1.2) involves
rectangles with sides parallel to the coordinate axis. Using the terminology
in [20] we may say that the scaling in (1.2) and in Theorem 1 is congruous
with the dependence axis of $X$. [20] showed that incongruous scaling of
linear LRD RF in (1.3) may dramatically change the scaling limits in (1.2) and
the scaling transition point $\gamma_{0}$. We expect that the spectrum
approach in our paper may lead to a comprehensive treatment of incongruous
scaling limits under various dependence assumptions.

## Acknowledgements

The author thanks Vytautė Pilipauskaitė for useful comments and Remigijus
Lapinskas for help with Figure 1 graphs.

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