Asymptotic Behaviour of Discretised Functionals of Long-Range Dependent Functional Data

1 Introduction

Recent advances in technology allowed collecting big data at high frequency (effectively continuous) rates that led to the ubiquity of functional data (samples of curves or surfaces) (Ramsay and Silverman (2005); Wang et al. (2016)). Handling such new complex data is essential in various applications, for example, in earth, environmental, ecological sciences, cosmology and image analysis. However, most of classical statistical models and results were developed for discretely sampled data.

Discretisation and corresponding additive models are often used as powerful dimension reduction tools in the analysis of functional data which are intrinsically infinite dimensional. The discretisation is a common strategy for approximating statistics of such data (see, for example, $\lx@sectionsign 6.4.3$ in Ramsay and Silverman (2005)). Also, in practice, the functional curves or surfaces are often observed only at a finite number of points.

Note, that various statistics of functional data can be expressed by integral functionals of these data or their transformations. For instance, some well-known examples of such statistics include sample moments and sample sojourn measures (Minkowski functionals) (see Leonenko and Olenko (2014)

). Another popular model in various applications (especially in engineering and signal processing) is stochastic processes that are obtained as outputs of filters, i.e. defined mathematically by a convolution integral operator. Another example is functional linear regression models defined by weighted integral functionals. These models found numerous statistical applications in medicine, linguistics, chemometrics (see

Ramsay and Silverman (2005); Crambes et al. (2009); Zhang (2014)).

In all above applications it is usually assumed that the discretisation error is negligible with respect to the estimation error. However, there are almost no known results that rigorously prove it. This paper addresses this problem and investigates discretisation errors for weighted functionals of long-range dependent spatial processes. Their rates of decay for the case of increasing observation windows are found. It is shown that both additive and integral functionals converge to the same limit distribution. It is proved that these distributions are non-Gaussian. These results provide a constructive method for determining the number of discretisation nodes for a given accuracy.

Various results in statistical inference of random fields were first obtained by Yadrenko (1983). Recently, considerable attention has been paid to asymptotic behaviour of non-linear statistics of random processes and fields (see Ivanov and Leonenko (2008); Bai and Taqqu (2013); Leonenko and Olenko (2014); Anh et al. (2019)

and the references therein). Direct probability techniques were used to study these statistics, for example, in regression models. Asymptotic distributions of these statistics were discussed by

Ivanov and Leonenko (1989) and it was shown that central and non-central limit theorems hold for particular models. However, no results about discretisations were given.

There are many practical situations in which non-Gaussian random processes and fields are appropriate for statistical modeling. We deal with an important class of models defined by non-linear functions of Gaussian random fields. This class is widely used in modeling non-Gaussian data. It can be analysed using Wiener chaos expansions that give good data approximations in many cases (see De Oliveira et al. (1997); Vio et al. (2001)).

This research deals with asymptotic behaviour of integral and additive non-linear functionals of random fields with long-range dependence. Long-range dependence is an empirical phenomenon which has been observed in different applied fields including cosmology, economics, geophysics, air pollution, image analysis, earth sciences, just to mention a few examples. For this reason, great effort has been devoted to studying models based on long-range dependent random fields (see Ivanov and Leonenko (1989); Wackernagel (1998); Doukhan et al. (2002); Frías et al. (2008)). Weighted functionals of long-range dependent random fields were considered in Olenko (2013); Ivanov et al. (2013); Ivanov and Leonenko (1989). These functionals can have non-Gaussian limits that are known as Hermite or Hermite-Rosenblatt distributions (Rosenblatt (1961); Taqqu (1975); Dobrushin and Major (1979); Taqqu (1979)

). Their asymptotic distributions can be characterised by either multiple Wiener-Itô integrals representations or characteristic functions (see

Taqqu (1979); Dobrushin and Major (1979); Leonenko and Taufer (2006)).

In various applications, it is natural to consider statistics of random fields and to study their limit behaviour over increasing spatial windows. In these cases, integrals of non-linear functionals of spatial functional data and additive non-linear functionals for discrete observations on a bounded region were studied in numerous papers (see, for example, Major (1981); Leonenko and Olenko (2014); Anh et al. (2015, 2019)). When we deal with the asymptotic behaviour of discretised functionals of functional data, it is important to know how asymptotics of these integrals are related to additive functionals. To the best of our knowledge only particular cases of this correspondence have been addressed in Leonenko and Taufer (2006) and Alodat and Olenko (2017) for rectangular observation windows. However, in many applications spatial data is not necessarily available over rectangles, but rather over irregularly-shaped regions (Cressie (1993); Lahiri et al. (1999)). Therefore, it is important to obtain theoretical results about asymptotics for general types of observation windows. In this paper we extend results of Leonenko and Taufer (2006) and Alodat and Olenko (2017) under more general conditions. More precisely, we consider weighted functionals of random fields of the form

d_{r}^{- 1} \int_{Δ_{n} (r)} g (x) H_{κ} (ξ (x)) d x, r \to \infty,

where $ξ (x), x \in R^{n}$ , is a long-range dependent random field, $g (x)$ and $H_{κ} (\cdot)$ are non-random functions, $Δ_{n} \subset R^{n}$ is an observation window and $d_{r}^{- 1}$ is a normalising factor. We show that these integrals and the corresponding discretised versions have same non-Gaussian limit distributions.

The article is organised as follows. In Section 2 we introduce main notations and outline necessary background from the theory of random fields. In Section 3 we recall some assumptions and auxiliary results from the spectral and correlation theory of random fields. In Section 4.1 we study the case of two-dimensional functionals. Section 4.2 gives a general multidimensional version of the results. Proofs are provided in Section 5. In Section 6 some simulations studies are presented to confirm theoretical results. Conclusions and directions for future research are presented in Section 7.

2 Definitions and Notations

This section provides basic definitions and notations that are used in this article.

In what follows $| \cdot |$ , $∥ \cdot ∥$ , $⌊ \cdot ⌋$ and $⌈ \cdot ⌉$ are used for the Lebesgue measure, the Euclidean distance in $R^{n}$ , the floor and ceiling functions, respectively. The symbols $C$ , $ε$ and $δ$ (with subscripts) will be used to denote constants that are not important for our discussion. Note, that the same symbol $C$ may be used for different constants appearing in the same proof. For a set $A \subset R^{n}, n \geq 1$ , we denote by $A^{\circ}$ and $A^{c}$ the interior and the exterior of the set $A$

respectively. Moreover, it is assumed that all random variables are defined on a fixed probability space

(Ω, F, P)

We consider a measurable mean-square continuous zero-mean homogeneous isotropic real-valued random field $ξ (x), x \in R^{n}$ , with the covariance function

B (r) = B (∥ x ∥) := E (ξ (0) ξ (x)), x \in R^{n} .

It is well known that there exists a bounded non-decreasing function $Φ (u), u ⩾ 0$ , (see Yadrenko (1983); Ivanov and Leonenko (1989)) such that

B (r) = (2 / r)^{(n - 2) / 2} Γ (n / 2) \int_{0}^{\infty} J_{(n - 2) / 2} (r u) u^{(2 - n) / 2} d Φ (u),

where $J_{v} (\cdot)$ is the Bessel function of the first kind of order $v > - 1 / 2$ .

The function $Φ (\cdot)$ is called the isotropic spectral measure of the random field $ξ (x),$ $x \in R^{n}$ . If there exists a function $φ (u), u \in [0, \infty)$ , such that

u^{n - 1} φ (u) \in L_{1} ([0, \infty)), Φ (u) = \frac{2 π^{n / 2}}{Γ (n / 2)} \int_{0}^{u} z^{n - 1} φ (z) d z,

then the function $φ (\cdot)$ is called the isotropic spectral density of the field $ξ (x)$ .

The field $ξ (x)$ with an absolutely continuous spectrum has the following isonormal spectral representation

ξ (x) = \int_{R^{n}} e^{i ⟨ λ, x ⟩} \sqrt{φ (∥ λ ∥)} W (d λ),

(2.1)

where $W (\cdot)$

is the complex Gaussian white noise random measure on

R^{n}

(see Yadrenko (1983); Ivanov and Leonenko (1989)).

The Hermite polynomials $H_{m} (x), m \geq 0$ , are defined by

H_{m} (x) := (- 1)^{m} exp (\frac{x^{2}}{2}) \frac{d^{m}}{d x^{m}} exp (- \frac{x^{2}}{2}) .

The first few Hermite polynomials are $H_{0} (x) = 1, H_{1} (x) = x, H_{2} (x) = x^{2} - 1, H_{3} (x) = x^{3} - 3 x .$

The Hermite polynomials $H_{m} (x), m \geq 0$ , form a complete orthogonal system in the Hilbert space $L_{2} (R, ϕ (ω) d ω) := {G : \int_{R} G^{2} (ω) ϕ (ω) d ω < \infty}$ , where $ϕ (ω)$

is the probability density function of the standard normal distribution. An arbitrary function

G (ω) \in L_{2} (R, ϕ (ω) d ω)

admits the mean-square convergent expansion

G (ω) = \infty \sum j = 0 \frac{C_{j} H_{j} (ω)}{j!}, C_{j} := \int_{R} G (ω) H_{j} (ω) ϕ (ω) d ω .

(2.2)

By Parseval’s identity it holds

\infty \sum j = 0 \frac{C_{j}^{2}}{j!} = \int_{R} G^{2} (ω) ϕ (ω) d ω .

Definition 2.1.

(Taqqu (1975)) Let $G (\cdot) \in L_{2} (R, ϕ (ω) d ω)$ . Assume that there exists an integer $κ ⩾ 1$ , such that $C_{j} = 0$ for all $0 < j \leq κ - 1$ , but $C_{κ} \neq 0$ . Then $κ$ is called the Hermite rank of $G (\cdot)$ and is denoted by $H r a n k G (\cdot) .$

Note, that by (2.1.8) in Ivanov and Leonenko (1989) we get $E (H_{m} (ξ (x))) = 0$ and

E (H_{m_{1}} (ξ (x)) H_{m_{2}} (ξ (y))) = δ_{m_{1}}^{m_{2}} m_{1}! B^{m_{1}} (∥ x - y ∥), x, y \in R^{n},

(2.3)

where $δ_{m_{1}}^{m_{2}}$ is the Kronecker delta function.

Definition 2.2.

(Bingham et al. (1989)) A measurable function $L : (0, \infty) \to (0, \infty)$ is slowly varying at infinity if for all $t > 0$ , ${lim}_{r \to \infty} L (t r) / L (r) = 1.$

3 Assumptions and Auxiliary Results

This section gives some assumptions and results from the spectral and correlation theory of random fields that will be used in the following sections.

Assumption 3.1.

Let $ξ (x), x \in R^{n}$ , be a homogeneous isotropic Gaussian random field with $E ξ (x) = 0$ and the covariance function $B (x)$ , such that $B (0) = 1$ and

B (x) = E (ξ (0) ξ (x)) = ∥ x ∥^{- α} L_{0} (∥ x ∥), α > 0,

where $L_{0} (∥ \cdot ∥)$ is a function slowly varying at infinity.

If $α \in (0, n)$ , then the covariance function $B (x)$ satisfying Assumption 3.1 is not integrable, which corresponds to the long-range dependence case (Anh et al. (2015)).

The notation $Δ_{n} \subset R^{n}$ will be used to denote a Jordan-measurable compact bounded set, such that $| Δ_{n} | > 0$ , and $Δ_{n}$ contains the origin in its interior. Let $Δ_{n} (r), r > 0$ , be the homothetic image of the set $Δ_{n}$ , with the centre of homothety at the origin and the coefficient $r > 0$ , that is $| Δ_{n} (r) | = r^{n} | Δ_{n} |$ and $Δ_{n} = Δ_{n} (1)$ . For $c \in R$ and $v \in R^{n}, n \geq 1$ , we define $cΔn:=\setcx:x∈Δn$ and $Δn−v:=\setx−v:x∈Δn$ .

Let $H r a n k G (\cdot) = κ$ . Denote the random variables $K_{r}$ and $K_{r, κ}$ by

K_{r} := \int_{Δ_{n} (r)} G (ξ (x)) d x and K_{r, κ} := \frac{C_{κ}}{κ!} \int_{Δ_{n} (r)} H_{κ} (ξ (x)) d x,

where $C_{κ}$ is given by (2.2).

Theorem 3.1.

(Leonenko and Olenko (2014)) Suppose that $ξ (x), x \in R^{n}$ , satisfies Assumption 3.1 and $H r a n k G (\cdot) = κ \geq 1$ . If a limit distribution exists for at least one of the random variables

\frac{K_{r}}{\sqrt{V a r K_{r}}} a n d \frac{K_{r, κ}}{\sqrt{V a r K_{r, κ}}},

then the limit distribution of the other random variable also exists, and the limit distributions coincide when $r \to \infty$ .

By Theorem 3.1 it is enough to study $K_{r, κ}$ to get asymptotic distributions of $K_{r}$ . Therefore, we restrict our attention only to $K_{r, κ}$ .

Assumption 3.2.

The random field $ξ (x), x \in R^{n}$ , has the isotropic spectral density

φ (∥ λ ∥) := c_{1} (n, α) ∥ λ ∥^{α - n} L (\frac{1}{∥ λ ∥}),

where $α \in (0, n), c_{1} (n, α) := Γ ((n - α) / 2) / 2^{α} π^{n / 2} Γ (α / 2),$ and $L (∥ \cdot ∥) \sim L_{0} (∥ \cdot ∥)$ is a locally bounded function which is slowly varying at infinity.

One can find more details on relations between Assumptions 3.1 and 3.2 in Anh et al. (2019).

The function $K_{Δ_{n}} (x)$

will be used to denote the Fourier transform of the indicator function of the set

Δ_{n}

, i.e.

K_{Δ_{n}} (x) := \int_{Δ_{n}} e^{i ⟨ u, x ⟩} d u, x \in R^{n} .

Theorem 3.2.

(Leonenko and Olenko (2014)) Let $ξ (x), x \in R^{n}$ , be a homogeneous isotropic Gaussian random field. If Assumptions 3.1 and 3.2 hold, $α \in (0, n / κ)$ , then for $r \to \infty$ the random variables

X_{r, κ} (Δ_{n}) := r^{κ α / 2 - n} L^{- κ / 2} (r) \int_{Δ_{n} (r)} H_{κ} (ξ (x)) d x

converge weakly to

X_{κ} (Δ_{n}) := c_{1}^{κ / 2} (n, α) \int_{R^{n κ}}^{'} K_{Δ_{n}} (λ_{1} + \dots + λ_{κ}) \frac{W (d λ_{1}) \dots W (d λ_{κ})}{∥ λ_{1} ∥^{(n - α) / 2} \dots ∥ λ_{κ} ∥^{(n - α) / 2}} .

Here $\int_{R^{n κ}}^{'}$

denotes the multiple Wiener-Itô integral with respect to a Gaussian white noise measure, where the diagonal hyperplanes

λ_{i} = \pm λ_{j}, i, j = 1, \dots, κ, i \neq j

, are excluded from the domain of integration.

Below we present a limit theorem and the corresponding assumptions on the weight function in the integral functionals from Ivanov and Leonenko (1989). These results will be generalised in the subsequent sections. The obtained results on asymptotic equivalence of additive and integral functionals of random fields will be used to obtain limit theorems for the case of discrete observations.

Assumption 3.3.

(Ivanov and Leonenko (1989)) Let $ϑ (x) = ϑ (∥ x ∥)$ be a radial continuous function that is positive for $∥ x ∥ > 0$ and such that for $α \in (0, n / κ)$

lim r \to \infty \int_{Δ_{n}} \int_{Δ_{n}} \frac{ϑ (r ∥ x ∥) ϑ (r ∥ y ∥) d x d y}{ϑ^{2} (r) ∥ x - y ∥^{α κ}} \in (0, \infty) .

Let $u (∥ λ ∥) := c_{1} (n, α) L (\frac{1}{∥ λ ∥})$ , where $L (\cdot)$ is from Assumption 3.2. In Section 2.10 in Ivanov and Leonenko (1989) the case when the function $u (∥ λ ∥)$ is continuous in a neighborhood of zero, bounded on $(0, \infty)$ and $u (0) \neq 0$ , was studied. It was assumed that there is a function $¯ ϑ (∥ x ∥)$ such that

∫Rnκκ∏j=1∥λj∥α−n\abs∫Δnei⟨λ1+⋯+λκ,x⟩¯ϑ(x)dx2κ∏j=1dλj<∞

and

lim r \to \infty \int_{R^{n κ}} ∣ ∣ ∣ \int_{Δ_{n}} e^{i ⟨ λ_{1} + \dots + λ_{κ}, x ⟩} (\frac{ϑ (r ∥ x ∥)}{ϑ (r)} κ \prod j = 1 \sqrt{\frac{u (∥ λ_{j} ∥ r^{- 1})}{u (0)}} - ¯ ϑ (x)) d x {∣ ∣ ∣}^{2} κ \prod j = 1 ∥ λ_{j} ∥^{α - n} κ \prod j = 1 d λ_{j} = 0.

Under these assumptions the following result was obtained.

Theorem 3.3.

(Ivanov and Leonenko (1989)) If Assumption 3.3 holds, then for $r \to \infty$ the random variables

Y_{r, κ} := \frac{1}{r^{n - κ α / 2} ϑ (r) u^{κ / 2} (0)} \int_{Δ_{n} (r)} ϑ (∥ x ∥) H_{κ} (ξ (x)) d x

converge weakly to

Y_{κ} := \int_{R^{n κ}}^{'} K_{Δ_{n}} (λ_{1} + \dots + λ_{κ}; ¯ ϑ) \frac{\prod_{j = 1}^{κ} W (d λ_{j})}{\prod_{j = 1}^{κ} ∥ λ_{j} ∥^{(n - α) / 2}},

where $α \in (0, min (\frac{n}{κ}, \frac{n + 1}{2}))$ and $K_{Δ_{n}} (λ; ¯ ϑ) := \int_{Δ_{n}} e^{i ⟨ λ, x ⟩} ¯ ϑ (x) d x .$

Note, that the above result is a specification of the results on convergence to stochastic processes in Ivanov and Leonenko (1989) where functionals over the observation windows $Δ_{n} (r t^{1 / n}), t \in [0, 1]$ , were studied. For simplicity, this paper deals with a particular case when $t = 1$ . But the results of this paper can be easily extended to the case of general $Δ_{n} (r t^{1 / n})$ and convergence to stochastic processes on $t \in [0, 1]$ .

4 Main Results

4.1 Two-dimensional Case

In this section we consider integrals of two-dimensional random fields over a bounded increasing observation window $Δ_{2} (r) \subset R^{2}, r > 0$ . We show that the limit distributions of these integrals and their corresponding additive functionals coincide.

Our setup is as follows. We assume that the set $Δ_{2}$ can be represented as

Δ_{2} = {(x, y) \in R^{2} : a \leq x \leq b, f_{l, 1} (x) \leq y \leq f_{u, 1} (x)},

where $a = min (x, y) \in Δ_{2} x, b = max (x, y) \in Δ_{2} x$ , $f_{l, 1} (x) < f_{u, 1} (x), x \in (a, b)$ , and $f_{q, 1} (x), q \in {l, u}$ , are smooth functions (i.e. $f_{q, 1} (x) \in C^{1}, q \in {l, u}$ , where $C^{1}$ is a class of functions with continuous first derivatives) except of the sets $M_{q} = {x_{1}^{q}, \dots, x_{k_{q}}^{q}} \subset [a, b]$ , where these functions have finite jumps. Here $k_{q}$ is the number of jumps of $f_{q, 1} (\cdot)$ , see, for example, Figure 0(a). That is $x_{j}^{q} \in M_{q}$ , $j = 1, \dots, k_{q}$ , are jump points of the functions $f_{q, 1} (\cdot), q \in {l, u},$ if $lim x \to x_{j}^{q +} f_{q, 1} (x) \neq lim x \to x_{j}^{q -} f_{q, 1} (x)$ but $lim x \to x_{j}^{q +} f_{q, 1} (x)$ and $lim x \to x_{j}^{q -} f_{q, 1} (x)$ both exist. Note that, by the homothety of $Δ_{2} (r), r > 0$ , the set $Δ_{2} (r)$ can be represented as $Δ_{2} (r) = {(x, y) \in R^{2} : a r \leq x \leq b r, f_{l, r} (x) \leq y \leq f_{u, r} (x)},$ where $f_{q, r} (x) = r f_{q, 1} (x / r), q \in {l, u}$ . As $Δ_{2}$ contains the origin in its interior, it follows that $a r < 0 < b r$ , $⌊ a r ⌋ \to - \infty$ , and $⌈ b r ⌉ \to \infty,$ as $r \to \infty$ .

Let $ξ (x, y), x, y \in R$ , be a real-valued homogeneous isotropic Gaussian random field satisfying Assumptions 3.1 and 3.1. We investigate the integrals

Y_{r, κ}^{(c)} := d_{r}^{- 1} \int_{Δ_{2} (r)} g (x, y) H_{κ} (ξ (x, y)) d x d y,

as $r \to \infty$ , where $g (x, y)$ is a non-random function such that $g (x, y) \neq 0$ when $x = y$ , and $d_{r}^{- 1}$ is a normalising factor.

We define the corresponding additive functional to $Y_{r, κ}^{(c)}, r > 0$ , by

Y_{r, κ}^{(d)}

:= d_{r}^{- 1} ⌈ b r ⌉ \sum i = ⌊ a r ⌋ f_{r}^{(u)} (i) \sum j = f_{r}^{(l)} (i) g (i, j) H_{κ} (ξ (i, j)),

(4.1)

where $f_{r}^{(l)} (i) := ⌊ inf x \in [i, i + 1) f_{l, r} (x) ⌋$ and $f_{r}^{(u)} (i) := ⌈ sup x \in [i, i + 1) f_{u, r} (x) ⌉$ , see Figure 0(b). It is assumed that

for a r \in (i, i + 1) : inf x \in [i, i + 1) f_{l, r} (x) = ⌊ inf x \in [a r, i + 1) f_{l, r} (x) ⌋, sup x \in [i, i + 1) f_{u, r} (x) = ⌈ sup x \in [a r, i + 1) f_{u, r} (x) ⌉;

for b r \in (i, i + 1) : inf x \in [i, i + 1) f_{l, r} (x) = ⌊ inf x \in (i, b r] f_{l, r} (x) ⌋, sup x \in [i, i + 1) f_{u, r} (x) = ⌈ sup x \in (i, b r] f_{u, r} (x) ⌉ .

Example 4.1.

Figure 1(a) visualises a realisation of the long-range dependent Cauchy field $ξ (x, y)$ over the set $Δ_{2}$ from Figure 0(a). This field satisfies Assumptions 3.1 and 3.1. The corresponding normal Q-Q plot of $Y_{r, κ}^{(d)}$ in Figure 1(b) was obtained by simulating $ξ (x, y)$ $1000$ times for the large value $r = 200$ and $g (x, y) \equiv 1$ . It is close to the asymptotic distribution and Figure 1(b)

shows its departure from the Gaussian distribution.

Remark 4.1.

The functionals $Y_{r, κ}^{(c)}$ and $Y_{r, κ}^{(d)}$ include various important statistics. For example, the case of $g (x, y) \equiv 1,$ $k = 1,$ and $H_{1} (t) = t$ corresponds to the sample mean estimator. High order sample moments can be expressed in terms of $Y_{r, κ}^{(c)}$ or $Y_{r, κ}^{(d)}$ by using the formula

t^{κ} = κ! ⌊ κ / 2 ⌋ \sum m = 0 \frac{1}{2^{m} m! (κ - 2 m)!} H_{κ - 2 m} (t) .

Another important example, a level excess measure, can be found by using the Hermite series expansion (2.2) for the indicator function $χ (t > C) :$

χ (t > C) = \infty \sum m = 0 \frac{C_{m}^{(C)} H_{m} (t)}{m!}, C_{m}^{(C)} = {\begin{matrix} 1 - Φ (C), & m = 0, ϕ (C) H_{m - 1} (C), & m \geq 1, \end{matrix}

where $Φ (\cdot)$ and $ϕ (\cdot)$ are the cdf and pdf for $N (0, 1)$ respectively. The case of $g (x, y) \equiv̸ 1$ corresponds to weighted versions of the above statistics.

Let us define rectangles $S_{q, r} (i), q \in {l, u}, i = ⌊ a r ⌋, \dots, ⌈ b r ⌉$ , as

S_{q, r} (i) := [i, i + 1) \times [inf x \in [i, i + 1) f_{q, r} (x), sup x \in [i, i + 1) f_{q, r} (x)] .

Then, $Y_{r, κ}^{(c)}$ can be rewritten as

	$Y_{r, κ}^{(c)}$	$= d_{r}^{- 1} \int_{A (r)} g (x, y) H_{κ} (ξ (x, y)) d x d y - d_{r}^{- 1} ⌈ b r ⌉ \sum i = ⌊ a r ⌋ \int_{S_{l, r} (i) \cap Δ_{2}^{c} (r)} g (x, y) H_{κ} (ξ (x, y)) d y d x$
		$- d_{r}^{- 1} ⌈ b r ⌉ \sum i = ⌊ a r ⌋ \int_{S_{u, r} (i) \cap Δ_{2}^{c} (r)} g (x, y) H_{κ} (ξ (x, y)) d y d x,$		(4.2)

where $A (r) := {(x, y) \in R^{2} : x \in [i, i + 1), y \in [f_{r}^{(l)} (i), f_{r}^{(u)} (i)], i = ⌊ a r ⌋, \dots, ⌈ b r ⌉}$ .

Assumption 4.1.

Let $g (u, v), u, v \in R$ , be such that $r^{4 - α κ} g^{2} (r, r) L^{κ} (r) \to \infty,$ as $r \to \infty,$ and there exists a function $g^{*} (u, v)$ such that for some $ε > 0$ uniformly for $(u, v) \in Δ_{2 ε} := Δ_{2} (1 + ε)$ it holds

limr→∞\absg(ru,rv)g(r,r)−g∗(u,v)→0,

where $α \in (0, 2 / κ)$ and

∫Δ2ε∫Δ2ε\absg∗(u1,u2)g∗(v1,v2)dv1dv2du1du2((u1−v1)2+(u2−v2)2)κα/2<∞.

Notice that, Assumption 4.1 is more general than Assumption 3.3.

Remark 4.2.

It follows from Assumption 4.1 that $g^{*} (u, v)$ is bounded on $Δ_{2 ε}$ .

Remark 4.3.

Note, that the conditions on the function $g (\cdot, \cdot)$ in Assumption 4.1 are met by numerous types of functions that are important in solving various statistical problems, in particular, non-linear regression and M estimators. For example, the functions

$g (u, v) = u^{μ_{1}} v^{μ_{2}} with g^{*} (u, v) = u^{μ_{1}} v^{μ_{2}},$
$g (u, v) = u v log (μ_{1} + u) log (μ_{2} + v) with g^{*} (u, v) = u v$

(for some appropriate constants $μ_{1}$ and $μ_{2}$ ) can be chosen. The case of $g (u, v) \equiv C > 0$ corresponds to the classical equally-weighted functionals and non-central limit theorems.

Remark 4.4.

To avoid degenerated cases, the condition $r^{4 - α κ} g^{2} (r, r) L^{κ} (r) \to \infty,$ as $r \to \infty,$

is essential to guarantee the boundedness of the variance of

d_{r}^{- 1} Y_{r, κ}^{(c)}

Now, we proceed to the main result.

Theorem 4.1.

Let $0 < α < 2 / κ$ . If Assumptions 3.1, 3.2 and 4.1 hold, then

lim \begin{matrix} r \to \infty \end{matrix} \frac{E {[\int_{Δ_{2} (r)} g (x, y) H_{κ} (ξ (x, y)) d y d x - \sum_{i = ⌊ a r ⌋}^{⌈ b r ⌉} \sum_{j = f_{r}^{(l)} (i)}^{f_{r}^{(u)} (i)} g (i, j) H_{κ} (ξ (i, j))]}_{Δ_{2} (r)}^{2}}{r^{4 - α κ} L^{κ} (r) g^{2} (r, r)} = 0.

(4.3)

Remark 4.5.

Theorem 4.1 is also true if $f_{q, 1} (\cdot), q \in {l, u}$ , are Lipschitz functions.

4.2 Multidimensional Case

This section gives a multidimensional version of Theorem 4.1 and a generalisation of Theorem 3.3. Then, we apply the obtained results to show that additive functionals have the same asymptotic distribution as the corresponding integral functionals.

We use the following notations that enable us to obtain the result of this section analogously to the two-dimensional case. Let $x := (x_{1}, \dots, x_{n}) \in R^{n}, 1_{n} := (1, \dots, 1) \in R^{n}$ , and the set $Δ_{n} \subset R^{n}, n \geq 3$ .

First, we consider the case of $n = 3$ . We assume that the set $Δ_{3}$ can be represented as

Δ_{3} = {(x_{1}, x_{2}, x_{3}) \in R^{3} : (x_{1}, x_{2}) \in Δ_{2}, f_{l, 1} (x_{1}, x_{2}) \leq x_{3} \leq f_{u, 1} (x_{1}, x_{2})},

where $f_{l, 1} (x_{1}, x_{2}) < f_{u, 1} (x_{1}, x_{2})$ , $(x_{1}, x_{2}) \in Δ_{2}^{\circ}$ , and $fq,1(x1,x2), q∈\setl,u$ , are smooth functions except of the sets where these functions have finite jumps, i.e.

M′q:=\set(x1,x2)∈Δ2:x1=xqj1 or x2=xqj2, j1=1,…,k(1)q, j2=1,…,k(2)q,

where $x_{j_{1}}^{q}$ and $x_{j_{2}}^{q}$ , are constants. Here $k_{q}^{(1)}$ and $k_{q}^{(2)}$ are the number of jumps of $f_{q, 1} (\cdot, \cdot)$ . Thus, $M_{q}^{'}$ consists of a finite number of two-dimensional line segments.

Note that, by the homothety of $Δ_{3} (r), r > 0$ , the set $Δ_{3} (r)$ can be represented as $Δ_{3} (r) = {(x_{1}, x_{2}, x_{3}) \in R^{3} : (x_{1}, x_{2}) \in Δ_{2} (r), f_{l, r} (x_{1}, x_{2}) \leq x_{3} \leq f_{u, r} (x_{1}, x_{2})}$ , where $f_{q, r} (x_{1}, x_{2}) = r f_{q, 1} (x_{1} / r, x_{2} / r)$ , $q∈\setl,u$ .

For a real-valued homogeneous isotropic Gaussian random field $ξ (x), x \in R^{3}$ , let

Z_{r, κ}^{(c)} := d_{r}^{- 1} \int_{Δ_{3} (r)} g (x) H_{κ} (ξ (x)) d x,

where $g (x), x \in R^{3}$ , is a non-random scalar function such that $g (x_{1} 1_{3}) \neq 0$ , and $d_{r}^{- 1}$ is a normalising factor.

We define the corresponding additive functional to $Z_{r, κ}^{(c)}, r > 0$ , by

Z_{r, κ}^{(d)}

:= d_{r}^{- 1} \sum (i_{1}, i_{2}, i_{3}) \in Q_{3} (Δ_{3} (r)) g (i_{1}, i_{2}, i_{3}) H_{κ} (ξ (i_{1}, i_{2}, i_{3})),

where

	$Q_{3} (Δ_{3} (r)) := {$	$(i1,i2,i3)∈Z3: i1∈\set⌊ar⌋,…,⌈br⌉, i2∈\setf(l)r(i1),…,f(u)r(i1),$
		$i3∈\setf(l)r(i1,i2),…,f(u)r(i1,i2)},$

and

	$P_{2} (r, i_{1}, i_{2}) := {$	$(x1,x2)∈[i1,i1+1)×[i2,i2+1]∩Δ2(r), i1∈\set⌊ar⌋,…,⌈br⌉,$
		$i_{2} \in [f_{r}^{(l)} (i_{1}), f_{r}^{(u)} (i_{1})]} .$

Remark 4.6.

The intersection with $Δ_{2} (r)$ is required to correctly define the infimum and supremum for the cases when some points in $[i_{1}, i_{1} + 1) \times [i_{2}, i_{2} + 1]$ are outside of $Δ_{2} (r)$ , which may happen for the boundary region.

For each $q \in {l, u}$ , define three-dimensional parallelepipeds $S_{q, r} (i_{1}, i_{2})$ as

S_{q, r} (i_{1}, i_{2}) := [i_{1}, i_{1} + 1) \times [i_{2}, i_{2} + 1] \times [inf \begin{matrix} (x_{1}, x_{2}) \in P_{2} (r, i_{1}, i_{2}) \end{matrix} f_{q, r} (x_{1}, x_{2}), sup \begin{matrix} (x_{1}, x_{2}) \in P_{2} (r, i_{1}, i_{2}) \end{matrix} f_{q, r} (x_{1}, x_{2})],

where $i_{1} = ⌊ a r ⌋, \dots, ⌈ b r ⌉, i$

Asymptotic Behaviour of Discretised Functionals of Long-Range Dependent Functional Data

On LSE in regression model for long-range dependent random fields on spheres

Limit theorems for filtered long-range dependent random fields

Reduction principle for functionals of strong-weak dependent vector random fields

Non-central limit theorems for functionals of random fields on hypersurfaces

Inference in Models of Discrete Choice with Social Interactions Using Network Data

Ordinal Patterns in Long-Range Dependent Time Series

A central limit theorem for the Benjamini-Hochberg false discovery proportion under a factor model

1 Introduction

2 Definitions and Notations

Definition 2.1.

Definition 2.2.

3 Assumptions and Auxiliary Results

Assumption 3.1.

Theorem 3.1.

Assumption 3.2.

Theorem 3.2.

Assumption 3.3.

Theorem 3.3.

4 Main Results

4.1 Two-dimensional Case

Example 4.1.

Remark 4.1.

Assumption 4.1.

Remark 4.2.

Remark 4.3.

Remark 4.4.

Theorem 4.1.

Remark 4.5.

4.2 Multidimensional Case

Remark 4.6.

Asymptotic Behaviour of Discretised Functionals of Long-Range Dependent Functional Data

Related Research

On LSE in regression model for long-range dependent random fields on spheres

Limit theorems for filtered long-range dependent random fields

Reduction principle for functionals of strong-weak dependent vector random fields

Non-central limit theorems for functionals of random fields on hypersurfaces

Inference in Models of Discrete Choice with Social Interactions Using Network Data

Ordinal Patterns in Long-Range Dependent Time Series

A central limit theorem for the Benjamini-Hochberg false discovery proportion under a factor model

1 Introduction

2 Definitions and Notations

Definition 2.1.

Definition 2.2.

3 Assumptions and Auxiliary Results

Assumption 3.1.

Theorem 3.1.

Assumption 3.2.

Theorem 3.2.

Assumption 3.3.

Theorem 3.3.

4 Main Results

4.1 Two-dimensional Case

Example 4.1.

Remark 4.1.

Assumption 4.1.

Remark 4.2.

Remark 4.3.

Remark 4.4.

Theorem 4.1.

Remark 4.5.

4.2 Multidimensional Case

Remark 4.6.