Fractional Operators and Fractionally Integrated Random Fields on Z

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Fractional Operators and Fractionally Integrated Random Fields on Z

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Donatas Surgailis^*,

Vytautė Pilipauskaitė

Donatas Surgailis^*,

Vytautė Pilipauskaitė

This version is not peer-reviewed

Submitted:

17 May 2024

Posted:

21 May 2024

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Abstract

We consider fractional integral operators $(I-T)^d, d \in (-1,1)$ acting on functions $g: \mathbb{Z}^{\nu} \to \mathbb{R}, \nu \ge 1 $, where $T $ is the transition operator of a random walk on $\mathbb{Z}^{\nu}$. We obtain sufficient and necessary conditions for the existence, invertibility and square summability of kernels $\tau (\mathbf{s}; d), \mathmb{s} \in \mathbb{Z}^{\nu}$ of $(I-T)^d $. Asymptotic behavior of $\tau (\mathbf{s}; d)$ as $|\mathbf{s}| \to \infty$ is identified following local limit theorem for random walk. A class of fractionally integrated random fields $X$ on $\mathbb{Z}^{\nu}$ solving the difference equation $(I-T)^d X = \varepsilon$ with white noise on the right-hand side is discussed, and their scaling limits. Several examples including fractional lattice Laplace and heat operators are studied in detail.

Keywords:

Subject: Computer Science and Mathematics - Probability and Statistics

1. Introduction

Classical fractional differentiation/integration operators

{(I - T)}^{d}, d \in (- 1, 1)

d \neq 0

, acting on functions

g : Z \to R

, where

(I - T) g (t) = g (t) - g (t - 1)

is `discrete derivative’ with respect to `time’

t \in Z

, are defined through the binomial expansion

{(1 - z)}^{d} = \sum_{j = 0}^{\infty} ψ_{j} (d) z^{j}, z \in C, | z | < 1

, viz.,

{(I - T)}^{d} g (t) : = \sum_{j = 0}^{\infty} ψ_{j} (d) T^{j} g (t) = \sum_{j = 0}^{\infty} ψ_{j} (d) g (t - j), t \in Z

(1)

with coefficients

ψ_{0} (d) : = 1

and

\begin{matrix} ψ_{j} (d) & : = & \frac{Γ (j - d)}{Γ (j + 1) Γ (- d)}, j \in N . \end{matrix}

(2)

Here

Γ

denotes the gamma function:

Γ (z) : = \int_{0}^{\infty} t^{z - 1} e^{- t} d t

z > 0

, and

Γ (z) : = \frac{1}{z} Γ (z + 1)

- 1 < z < 0

. The asymptotics

ψ_{j} (d) \sim Γ {(- d)}^{- 1} j^{- d - 1} (j \to \infty), 0 < | d | < 1

(3)

(which follows by application of Stirling’s formula to (2)) determines the class of functions g and summability properties of (1).

Fractional operators in (1) play important role in the theory of discrete-time stochastic processes, in particular, time series. See, e.g., the monographs [5,10,14,25,31] and the references therein. The autoregressive fractionally integrated moving-average ARFIMA

(0, d, 0)

process

{X (t); t \in Z}

is defined as a stationary solution of the stochastic difference equation

{(I - T)}^{d} X (t) = \sum_{j = 0}^{\infty} ψ_{j} (d) X (t - j) = ε (t), t \in Z

(4)

with white noise (a sequence of standardized uncorrelated random variables (r.v.s))

{ε (t); t \in Z}

. For

d \in (- 1 / 2, 1 / 2)

the solution of (4) is obtained by applying the inverse operator, viz.,

X (t) = {(I - T)}^{- d} ε (t) = \sum_{j = 0}^{\infty} ψ_{j} (- d) ε (t - j), t \in Z .

(5)

Since (3) implies

\sum_{j = 0}^{\infty} ψ_{j} {(d)}^{2} < \infty

(

| d | < 1 / 2

), (5) is a well-defined stationary process with zero mean and finite variance. ARFIMA

(0, d, 0)

process is the basic parametric model in statistical inference for long memory processes (also referred to as processes with long-range dependence). It has an explicit covariance function and the spectral density

\begin{matrix} f (x) & = & {(2 π)}^{- 1} {| 1 - e^{- ß x} |}^{- 2 d}, x \in Π : = [- π, π] \end{matrix}

(6)

which explodes or vanishes at the origin

x = 0

{(2 π)}^{- 1} {| x |}^{- 2 d}

depending on the sign of d.

In this paper, we extend fractional operators in (1) to functions g on

ν

-dimensional lattice

Z^{ν}, ν \geq 1

and/or more general T of the form

\begin{matrix} T g (t) & = & \sum_{u \in Z^{ν}} g (t + u) p (u) = E g (S_{1} + t), t \in Z^{ν}, \end{matrix}

(7)

where

{S_{j}; j \geq 0}

is a random walk on

Z^{ν}

starting at

S_{0} = 0

, with (1-step) probabilities

p = {p (u) : = P (S_{1} = u); u \in Z^{ν}}

. We assume that

p (0) < 1

, i.e. the random walk is non-degenerate at

0

. Clearly,

T^{j} g (t) = \sum_{u \in Z^{ν}} g (t + u) p_{j} (u) = E g (S_{j} + t), t \in Z^{ν}

, where

p_{0} (u) = I (u = 0), p_{j} (u) : = P (S_{j} = u),

u \in Z^{ν}

are the j-step probabilities,

j = 0, 1, 2, \dots

. Similarly to (4), we define fractional operators

{(I - T)}^{d}, - 1 < d < 1, d \neq 0

acting on

g : Z^{ν} \to R

\begin{matrix} {(I - T)}^{d} g (t) & = & \sum_{j = 0}^{\infty} ψ_{j} (d) T^{j} g (t) = \sum_{u \in Z^{ν}} τ (u; d) g (t + u), t \in Z^{ν} \end{matrix}

(8)

with coefficients

\begin{matrix} τ (u; d) & : = & \sum_{j = 0}^{\infty} ψ_{j} (d) p_{j} (u), \end{matrix}

(9)

expressed through the binomial coefficients

ψ_{j} (d)

and random walk probabilities

p_{j} (u)

Let us describe the content and results of this paper in more detail. The main result of Section 2 is Theorem 1 which provides the sufficient condition

\begin{matrix} \int_{Π^{ν}} {| 1 - \hat{p} (x) |}^{- 2 | d |} \underset{\cdot}{x} < \infty \end{matrix}

(10)

for invertibility

{(I - T)}^{d} {(I - T)}^{- d} = I

and square summability of fractional coefficients in (9), in terms of the characteristic function

\hat{p} (x) : = E exp {ß i 〈 x, S_{1} 〉}

(the Fourier transform) of the random walk. Section 2 also includes a discussion of the asymptotics of (9) as

| u | \to \infty

which is important in limit theorems and other applications of of fractional integrated random fields. Using classical local limit theorems, Propositions 1 and 2 obtain `isotropic’ asymptotics of (9) for a large class of random walk

{S_{j}}

showing that

τ (u; d)

decay as

O (| u |^{- ν - 2 d})

, hence,

\sum_{u \in Z^{ν}} | τ (u; - d) | = \infty (d > 0)

. The last fact is interpreted as long-range dependence [14,25,30] of the fractionally integrated random field

{X (t); t \in Z^{ν}}

defined as a stationary solution of the difference equation

{(I - T)}^{d} X (t) = ε (t), t \in Z^{ν}

(11)

with white noise on the r.h.s. and studied in studied in Section 3. Corollary 1 obtains conditions for the existence of stationary solution of (11) given by the inverse operator

X (t) = {(I - T)}^{- d} ε (t)

which are detailed in Examples 1 and 2 for fractional Laplacian and fractional heat operators. Section 2 and 3 also include a discussion of tempered fractional operators

{(I - r T)}^{d}, r \in (0, 1)

and tempered fractional random fields solving analogous equation

{(I - r T)}^{d} X (t) = ε (t)

which generalize the class of tempered ARFIMA processes [29] and have short-range dependence and a summable covariance function.

Section 4 is devoted to scaling limits of moving average random fields on

Z^{ν}

with coefficients satisfying Assumption (A)

(d)

which includes `isotropic’ fractional coefficients

τ (u; - d)

as a special case. The scaling limits refer to integrals

X_{λ} (ϕ) = \int_{R^{ν}} X ([t]) ϕ (t / λ) \underset{\cdot}{t}

of random field

{X (t); t \in Z^{ν}}

for each

ϕ : R^{ν} \to R

from a class of (test) functions, as scaling parameter

λ \to \infty

. The scaling limits are identified in Corollary 3 as self-similar Gaussian random fields with Hurst parameter

H = \frac{ν - 4 d}{2}

. We note that limit theorems for random fields with long-range dependence or negative dependence were studied in many works [7,8,9,18,22,23,27,28,32,33], including statistical applications [2,10,11,17].

We expect that this study can be extended in several directions, including anisotropic scaling, infinite variance random fields, and fractional operators in

R^{ν}

. See [1,6,16,20,21,24] for discussion and properties of fractional random fields with continuous argument

t \in R^{ν}

Notation. In what follows, C denote generic positive constants which may be different at different locations. We write

\begin{matrix} \overset{d}{⟶} \end{matrix}

and

\begin{matrix} \overset{d}{=} \end{matrix}

for the weak convergence and equality of probability distributions. Denote by

| \cdot |

the absolute-value norm on

K

, where

K

is either

R

C

, and the Euclidean norm on

R^{ν}

〈 \cdot, \cdot 〉

is the scalar product in

R^{ν}

. Denote by

e_{j}

the vector in

R^{ν}

with 1 in the jth coordinate and 0’s elsewhere. For

p \geq 1

, denote by

L^{p} (Z^{ν})

the space of functions

f : Z^{ν} \to K

, for which

\sum_{u \in Z^{ν}} {| f (u) |}^{p} < \infty

, and by

L^{p} (R^{ν})

the space of measurable functions

f : R^{ν} \mapsto K

, for which the p-th power of the absolute value is integrable with respect to the Lebesgue measure:

{∥ f ∥}_{L^{p} (R^{ν})} : = (\int_{R^{ν}} {| f (x) |}^{p} {\underset{\cdot}{x})}^{1 / p} < \infty

, with identification of functions

f, g

such that

f = g

almost everywhere (a.e.). Denote by

L^{\infty} (R^{ν})

the space of measurable and functions

f : R^{ν} \to K

, for which

{∥ f ∥}_{L^{\infty} (R^{ν})} : = inf {C \geq 0 : | f | \leq C a . e .} < \infty

, with identification of functions

f, g

such that

f = g

a.e. Write

I

for the indicator function. Write

[x]

for the smallest integer greater than or equal to

x \in R

i : = \sqrt{- 1} \in C, Z_{0}^{ν} : = Z^{ν} ∖ {0}

2. Invertibility and Properties of Fractional Operators

We start with properties of binomial coefficients in (2)

\begin{matrix} ψ_{j} (d) & < & 0 (j \geq 1), \sum_{j = 0}^{\infty} ψ_{j} (d) = 0 if 0 < d < 1, \\ ψ_{j} (d) & > & 0 (j \geq 1), \sum_{j = 0}^{\infty} ψ_{j} (d) = \infty if - 1 < d < 0 . \end{matrix}

(12)

The identity

{(1 - z)}^{d} {(1 - z)}^{- d} = 1

leads to

\begin{matrix} 1 & = & \sum_{j, k = 0}^{\infty} ψ_{j} (d) ψ_{k} (- d) z^{j + k} = \sum_{n = 0}^{\infty} z^{n} \sum_{j = 0}^{n} ψ_{j} (d) ψ_{n - j} (- d) \end{matrix}

and the invertibility relation:

\begin{matrix} \sum_{j = 0}^{n} ψ_{j} (d) ψ_{n - j} (- d) = I (n = 0), n \geq 0 . \end{matrix}

(13)

The following lemma gives some basic properties of fractional coefficients

τ (u; d)

in (9).

Lemma 1.

(i) Let

0 < d < 1

. Then the series in (9) converges for every

u \in Z^{ν}

and

\begin{matrix} τ (0; d) > 0, τ (u; d) \leq 0 (u \neq 0) and \sum_{u \in Z^{ν}} τ (u; d) = 0 . \end{matrix}

(14)

(ii) Let

- 1 < d < 0

. Then

0 \leq τ (u; d) \leq \infty

for every

u \in Z^{ν}

and

τ (0; d) \geq 1

and

\sum_{u \in Z^{ν}} τ (u; d) = \infty .

Moreover,

τ (0; d) < \infty

implies

τ (u; d) < \infty

and

\begin{matrix} - \sum_{u \neq 0} τ (u; d) τ (- u; - d) & \leq & τ (0; d) < \infty . \end{matrix}

(15)

(iii) Let

0 < d < 1

and

τ (0; - d) < \infty

. Then

\begin{matrix} \sum_{s \in Z^{ν}} τ (s; d) τ (t - s; - d) & = & I (t = 0), t \in Z^{ν} . \end{matrix}

(16)

Proof.

(i) From (9) and (12) we get

\begin{matrix} τ (0; d) & = & 1 + \sum_{j = 1}^{\infty} ψ_{j} (d) p_{j} (0) > 1 + \sum_{j = 1}^{\infty} ψ_{j} (d) = 0 \end{matrix}

(17)

since

p_{j} (0) = 1 (\forall j \geq 1)

is not possible. On the other hand, for

u \neq 0

we have

p_{0} (u) = 0

and

\begin{matrix} τ (u; d) & = & \sum_{j = 1}^{\infty} ψ_{j} (d) p_{j} (u) \leq 0 \end{matrix}

(18)

in view of (12).

(ii) Since

ψ_{j} (d) p_{j} (u) \geq 0

is obvious from (12), it suffices to show (15) since it implies

τ (u; d) < \infty

by (14). We have

\begin{matrix} Σ_{0} : = \sum_{u \neq 0} τ (u; d) (- τ (- u; - d)) & = & \sum_{u \neq 0} \sum_{j, k = 1}^{\infty} ψ_{j} (d) (- ψ_{k} (- d)) p_{j} (u) p_{k} (- u) \\ = & \sum_{n = 2}^{\infty} \sum_{j = 1}^{n - 1} ψ_{j} (d) (- ψ_{n - j} (- d)) \sum_{u \neq 0} p_{j} (u) p_{n - j} (- u) \end{matrix}

where exchanging the order of summation is legitimate as all summands are nonnegative. Hence, using

\sum_{u \neq 0} p_{j} (u) p_{n - j} (- u) \leq p_{n} (0)

and (13), we get

\begin{matrix} Σ_{0} & \leq & \sum_{n = 2}^{\infty} p_{n} (0) \sum_{j = 1}^{n - 1} ψ_{j} (d) (- ψ_{n - j} (- d)) = \sum_{n = 2}^{\infty} p_{n} (0) (ψ_{n} (d) + ψ_{n} (- d)) \\ \leq & \sum_{n = 2}^{\infty} p_{n} (0) ψ_{n} (d) < τ (0; d) \end{matrix}

proving part (ii).

(iii) The convergence of the series in (16) and the equality follow as in (15):

\begin{matrix} \sum_{s \in Z^{ν}} τ (s; d) τ (t - s; - d) & = & \sum_{j, k = 0}^{\infty} ψ_{j} (d) ψ_{k} (- d) \sum_{s \in Z^{ν}} p_{j} (s) p_{k} (t - s) \\ = & \sum_{n = 0}^{\infty} p_{n} (t) \sum_{j = 0}^{n} ψ_{j} (d) ψ_{n - j} (- d) \\ = & p_{0} (t) = I (t = 0) . \end{matrix}

Lemma 1 is proved. □

Remark 1.

Let

0 < d < 1

. Then the inequalities are strict:

τ (u; d) < 0

and

τ (u; - d) > 0

, if

p_{j} (u) > 0

for some j, i.e.

u

is accessible from state

0

. Moreover, if state

0

is transient, i.e. the probability of eventual return to

0

is strictly less that 1, which is equivalent to

\sum_{j = 0}^{\infty} p_{j} (0) < \infty

, then

τ (0; - d) < \infty

The main result of this section is Theorem 1, which provides necessary and sufficient conditions for square summability of fractional coefficients in (9), in terms of the characteristic function

\hat{p} (x)

, see (10). Write

\hat{f}

for the Fourier transform of a function

f : Z^{ν} \to R

. For

r \in (0, 1), d \in (- 1, 1)

introduce the tempered fractional operators

\begin{matrix} {(I - r T)}^{d} g (t) & = & \sum_{j = 0}^{\infty} r^{j} ψ_{j} (d) T^{j} g (t) = \sum_{u \in Z^{ν}} τ_{r} (u; d) g (t + u), t \in Z^{ν} \end{matrix}

(19)

with coefficients

\begin{matrix} τ_{r} (u; d) & : = & \sum_{j = 0}^{\infty} r^{j} ψ_{j} (d) p_{j} (u), \end{matrix}

(20)

and the Fourier transform

{\hat{τ}}_{r} (x; d) = {(1 - r \hat{p} (x))}^{d} .

Theorem 1.

For

- 1 < d < 1

, the following conditions are equivalent:

\begin{matrix} \int_{Π^{ν}} {| 1 - \hat{p} (x) |}^{- 2 | d |} \underset{\cdot}{x} & < & \infty, \end{matrix}

(21)

\begin{matrix} \sum_{u \in Z^{ν}} τ {(u; - | d |)}^{2} & < & \infty . \end{matrix}

(22)

Either of these conditions implies

\begin{matrix} \hat{τ} (\cdot; - | d |) & = & {(1 - \hat{p} (\cdot))}^{- | d |} in L^{2} (Π^{ν}) . \end{matrix}

(23)

Moreover, for

0 < d < 1

, the above conditions (21), (22) and (23) hold with d in place of

- | d |

Proof.

Let

0 < d < 1

. Firstly, we consider

τ (u; d)

in (9). They satisfy

\sum_{u \in Z^{ν}} | τ (u; d) | \leq \sum_{j = 0}^{\infty} | ψ_{j} (d) | < \infty

because of (3) and

\sum_{u \in Z^{ν}} p_{j} (u) = 1

with

0 \leq p_{j} (u) \leq 1

. Then

\sum_{u \in Z^{ν}} τ {(u; d)}^{2} < \infty

is immediate. Moreover, we have the Fourier transform

\hat{τ} (x; d) = \sum_{j = 0}^{\infty} ψ_{j} (d) {\hat{p}}_{j} (x)

, where

{\hat{p}}_{j} (x) = \hat{p} {(x)}^{j}

satisfies

| \hat{p} (x) | \leq 1

. We see that

\begin{matrix} \hat{τ} (x; d) = {(1 - \hat{p} (x))}^{d}, x \in Π^{ν}, \end{matrix}

(24)

belongs to

L^{2} (Π^{ν})

Now let us prove the implication (21) ⇒ (22). We use approximation by tempered fractional coefficients

τ_{r} (u; - d)

in (20) as

r ↗ 1

. We have that

{\hat{τ}}_{r} (x; - d) = {(1 - r \hat{p} (x))}^{- d} \to {(1 - \hat{p} (x))}^{- d}

a.e. as

r ↗ 1

. Next, for

z \in C

| z | \leq 1

0 < r < 1

, the inequality

| 1 - z | \leq | 1 - r z | + | r z - z | \leq | 1 - r z | + 1 - r

, where

1 - r \leq 1 - | r z | \leq | 1 - r z |

, becomes

| 1 - z | \leq 2 | 1 - r z |

. Using it we get the domination for all

0 < r < 1

x \in Π^{ν}

| {\hat{τ}}_{r} (x; - d) | \leq \frac{1}{| 1 - r \hat{p} {(x) |}^{d}} \leq \frac{2^{d}}{| 1 - \hat{p} {(x) |}^{d}}

by a function in

L^{2} (Π^{ν})

according to (21). Hence, by the dominated convergence theorem (DCT),

{\hat{τ}}_{r} (\cdot; - d) \to {(1 - \hat{p} (\cdot))}^{- d}

r ↗ 1

holds in

L^{2} (Π^{ν})

. As a consequence,

{\hat{τ}}_{r} (\cdot; - d)

0 < r < 1

, is a Cauchy sequence in

L^{2} (Π^{ν})

. By Parseval’s theorem, the inverse Fourier transforms

τ_{r} (u; - d) = \frac{1}{{(2 π)}^{ν}} \int_{Π^{ν}} e^{- 〈 u, x 〉} {\hat{τ}}_{r} (x; - d) \underset{\cdot}{x}, u \in Z^{ν}, 0 < r < 1,

is a Cauchy sequence in

L^{2} (Z^{ν})

and so

τ_{r} (\cdot; - d)

converges in

L^{2} (Z^{ν})

to some

f \in L^{2} (Z^{ν})

r ↗ 1

. This f must be

τ (\cdot; - d)

because

τ_{r} (u; - d) ↗ τ (u; - d)

r ↗ 1

for all

u

. We conclude that

τ (\cdot; - d) \in L^{2} (Z^{ν})

, or (22).

Let us turn to the implication (22) ⇒ (21). From (22) and

τ_{r} (u; - d) ↗ τ (u; - d)

for all

u

it follows that

τ_{r} (\cdot; - d) \to τ (\cdot; - d)

r ↗ 1

holds in

L^{2} (Z^{ν})

. By Parseval’s theorem,

{\hat{τ}}_{r} (\cdot; - d) = {(1 - r \hat{p} (\cdot))}^{- d}

0 < r < 1

, is a Cauchy sequence in

L^{2} (Π^{ν})

. It follows that

{lim}_{r ↗ 1} \int_{Π^{ν}} {| {\hat{τ}}_{r} (x; - d) - g (x) |}^{2} \underset{\cdot}{x} = 0

for some

g \in L^{2} (Π^{ν})

. We also have that

{lim}_{r ↗ 1} {(1 - r \hat{p} (x))}^{- d} = {(1 - \hat{p} (x))}^{- d}

for each

x \in Π^{ν}

such that

\hat{p} (x) \neq 1

. Since

{Leb}_{ν} (x \in Π^{ν} : \hat{p} (x) = 1) = 0

, see Lemma 2.3.2(a) in [19], we conclude that

g (\cdot) = {(1 - \hat{p} (\cdot))}^{- d}

a.e., proving (21).

The above argument also proves (23). On one hand,

\hat{τ} (\cdot; - d)

is the limit of

{\hat{τ}}_{r} (\cdot; - d)

L^{2} (Π^{ν})

r ↗ 1

because

τ_{r} (\cdot; - d)

converges in

L^{2} (Z^{ν})

τ (\cdot; - d)

r ↗ 1

. On the other hand,

{\hat{τ}}_{r} (\cdot; - d) = {(1 - r \hat{p} (\cdot))}^{- d} \to {(1 - \hat{p} (\cdot))}^{- d}

L^{2} (Π^{ν})

r ↗ 1

. We conclude that

\hat{τ} (\cdot; - d) = (1 - \hat{p} (\cdot))

a.e. Theorem 1 is proved. □

Next, we turn to asymptotics of `fractional coefficients’

τ (u; d)

in (9). The proof uses the local limit theorem in [19] for random walk probabilities

p_{j} (u) = P (S_{j} = u)

. Following the latter work we assume that

E e^{c | S_{1} |} < \infty (\exists c > 0) and {S_{j}} is zero mean, aperiodic, irreducible .

(25)

Conditions in (25) imply that the random walk has zero mean:

E S_{1} = \sum_{u \in Z^{ν}} u p (u) = 0

and invertible covariance matrix

Γ : = E S_{1} S_{1}^{'} .

(26)

According to the classical (integral) CLT, the normalized sum

S_{j} / \sqrt{j}, j \to \infty

approaches a Gaussian distribution on

R^{ν}

with density

\begin{matrix} ϕ (z) : = \frac{1}{{(2 π)}^{ν / 2} \sqrt{\det Γ}} e^{- 〈 z, Γ^{- 1} z 〉 / 2}, z \in R^{ν} . \end{matrix}

(27)

Denote

\begin{matrix} {\bar{p}}_{j} (u) : = \frac{1}{{(2 π j)}^{ν / 2} \sqrt{\det Γ}} e^{- 〈 u, Γ^{- 1} u 〉 / 2 j}, u \in R^{ν} . \end{matrix}

(28)

Lemma 2.

[19] Under conditions (25) there exists

C > 0

such that

| p_{j} (u) - {\bar{p}}_{j} (u) | \leq C {\bar{p}}_{j} (u) (\frac{1}{j^{1 / 2}} + \frac{{| u |}^{3}}{j^{2}}), \forall | u | < j^{2}, u \in Z^{ν} .

(29)

For `very atypical’ values

| S_{j} | > j

we use the following bound [19]: for any

k \geq 1

there exists

C > 0

such that

\begin{matrix} P (| S_{j} | > z \sqrt{j}) \leq C z^{- k}, \forall z > 0 . \end{matrix}

(30)

Proposition 1.

Let

p = {p (u); u \in Z^{ν}}

satisfy (25). The coefficients in (9) are well-defined for any

- (1 \land \frac{ν}{2}) < d < 1, d \neq 0

and satisfy

τ (u; d) = (B_{1} (d) + o (1)) {〈 u, Γ^{- 1} u 〉}^{- (ν / 2) - d}, | u | \to \infty,

(31)

where

\begin{matrix} B_{1} (d) : = \frac{2^{d} Γ (d + (ν / 2))}{π^{ν / 2} Γ (- d) \sqrt{\det Γ}} . \end{matrix}

Proof.

Let us prove (31). Since

Γ

is positive-definite,

{| u |}_{Γ} : = \sqrt{〈 u, Γ^{- 1} u 〉}

u \in R^{ν}

, is a norm. Note that it is equivalent to the Euclidean norm. Using (9) for a large

K > 0

decompose

{| u |}_{Γ}^{ν + 2 d} τ (u; d) = \sum_{i = 1}^{3} J_{i} (u)

, where

\begin{matrix} J_{1} (u) & : = & {| u |}_{Γ}^{ν + 2 d} Γ {(- d)}^{- 1} \sum_{{j > | u |}_{Γ}^{2} / K} j^{- d - 1} p_{j} (u), \\ J_{2} (u) & : = & {| u |}_{Γ}^{ν + 2 d} \sum_{{j > | u |}_{Γ}^{2} / K} (ψ_{j} (d) - Γ {(- d)}^{- 1} j^{- d - 1}) p_{j} (u), \\ J_{3} (u) & : = & {| u |}_{Γ}^{ν + 2 d} \sum_{0 \leq j \leq {| u |}_{Γ}^{2} / K} ψ_{j} (d) p_{j} (u) . \end{matrix}

It suffices to show that

\begin{matrix} lim_{K \to \infty} lim_{| u | \to \infty} J_{1} (u) & = & B_{1} (d), \\ lim_{K \to \infty} \underset{| u | \to \infty}{lim sup} J_{i} (u) & = & 0, i = 2, 3 . \end{matrix}

(32)

To show the first relation in (32) use (29). We have

J_{1} (u) = J_{1}^{'} (u) + J_{1}^{''} (u)

, where, for each

K > 0

fixed, the main term

J_{1}^{'} (u)

and the remainder term

J_{1}^{''} (u)

asymptotically behave when

| u | \to \infty

\begin{matrix} J_{1}^{'} (u) & : = & {| u |}_{Γ}^{ν + 2 d} Γ {(- d)}^{- 1} \sum_{{j > | u |}_{Γ}^{2} / K} j^{- d - 1} {\bar{p}}_{j} (u) \\ = & \frac{{| u |}_{Γ}^{ν + 2 d}}{{(2 π)}^{ν / 2} Γ (- d) \sqrt{\det Γ}} \int_{0}^{\infty} {I (| u |}_{Γ}^{2} {/ K < [y]) [y]}^{- d - 1 - (ν / 2)} e^{- {| u |}_{Γ}^{2} / 2 [y]} \underset{\cdot}{y} \\ \sim & \frac{1}{{(2 π)}^{ν / 2} Γ (- d) \sqrt{\det Γ}} \int_{1 / K}^{\infty} x^{- d - 1 - (ν / 2)} e^{- 1 / 2 x} \underset{\cdot}{x} \end{matrix}

and, for some constants

C, c > 0

\begin{matrix} | J_{1}^{''} (u) | & \leq & {C | u |}_{Γ}^{ν + 2 d} K^{3 / 2} \sum_{{j > | u |}_{Γ}^{2} / K} j^{- d - 3 / 2} {\bar{p}}_{j} (u) \\ \leq & {C | u |}_{Γ}^{- 1} K^{3 / 2} \int_{0}^{\infty} x^{- d - (3 / 2) - (ν / 2)} e^{- c / x} \underset{\cdot}{x} = o (1) . \end{matrix}

Hence, the first relation in (32) follows using

\int_{0}^{\infty} x^{- 1 - τ} e^{- 1 / x} d x = Γ (τ), τ > 0

. In view of (3), the same argument also proves the second relation in (32) for

i = 2

Consider (32) for

i = 3

. Split

J_{3} (u) = J_{3}^{''} (u) + J_{3}^{'} (u)

into two sums over

j > 0

, where

j^{2} \leq | u |

and

j^{2} > | u |

respectively. In the sum

J_{3}^{'} (u)

we also have that

j \leq {| u |}_{Γ}^{2} / K \leq {| u |}^{2}

and Lemma 2 entails the bound

p_{j} (u) \leq C {\bar{p}}_{j} (u) (\frac{{| u |}^{3}}{j^{2}}) \leq C {| u |}^{3} j^{- (ν / 2) - 2} e^{- {c | u |}^{2} / j}

(33)

for some constants

C, c > 0

. Hence,

\begin{matrix} | J_{3}^{'} (u) | & \leq & {C | u |}^{ν + 2 d + 3} \int_{0}^{{| u |}^{2}} {[y]}^{- d - 3 - (ν / 2)} e^{- {c | u |}^{2} / [y]} d y \\ \leq & {C | u |}^{- 1} \int_{0}^{1} x^{- d - 3 - (ν / 2)} e^{- c / x} \underset{\cdot}{x} = o (1) \end{matrix}

since the last integral converges for any d. Finally, by (30), given large enough

k > 0

, there exists

C > 0

such that

p_{j} (u) \leq C j^{k / 2} / {| u |}^{k}

, which implies

J_{3}^{''} (u) = o (1)

. This proves (32) and completes the proof of Proposition 1. □

Lemma 2 does not apply to the simple random walk (which is not aperiodic) in which case the local CLT takes a somewhat different form, see [19]. The application of the latter result and the argument in the proof of Proposition 1 yields the following result.

Proposition 2.

Let

p (e_{j}) = p (- e_{j}) = \frac{1}{2 ν}

j = 1, \dots, ν

. The coefficients in (9) are well-defined for any

- (1 \land \frac{ν}{2}) < d < 1, d \neq 0

and satisfy

τ (u; d) = {(B (d) + o (1)) | u |}^{- ν - 2 d}, | u | \to \infty,

(34)

where

\begin{matrix} B (d) : = \frac{2^{d} Γ (d + (ν / 2))}{ν^{d} Γ (- d)} . \end{matrix}

(35)

Proposition 1 and as well as Lemma 2 do not apply to random walks with non-zero mean as in Example 2 below (fractional heat operator) in which case fractional coefficients exhibit an anisotropic behavior different from (31). Such behavior is described in the following proposition. We assume that the underlying random walk factorizes into a deterministic drift by 1 in direction

- e_{1}

and a random walk on

Z^{ν - 1}

as in Lemma 2:

p (u) = \{\begin{matrix} 1 - θ, & u = - e_{1}, \\ θ \tilde{q} (\tilde{u}), & u = - e_{1} + (0, \tilde{u}), \end{matrix}

(36)

where

θ \in (0, 1)

and

\tilde{q} (\tilde{u})

is a probability distribution concentrated on

u = (u_{2}, \dots, u_{ν}) \in Z^{ν - 1}

such that

\tilde{u} \neq 0

. Write

{S_{j}; j \geq 0}

for the random walk starting at

0

with j-step probabilities

P ({\tilde{S}}_{j} = \tilde{u} | {\tilde{S}}_{0} = 0) = : {\tilde{q}}_{j} (\tilde{u})

j = 0, 1, \dots

, such that

{\tilde{q}}_{1} (\tilde{u}) : = \tilde{q} (\tilde{u})

\tilde{u} \in Z^{ν - 1}

. In order to apply Lemma 2, we make a similar assumption to (25):

E e^{c | {\tilde{S}}_{1} |} < \infty (\exists c > 0) and {{\tilde{S}}_{j}} is zero mean, irreducible

(37)

and denote

\tilde{Γ} : = E {\tilde{S}}_{1} {\tilde{S}}_{1}^{'}

the respective covariance matrix. Let

ρ (x) : = {(x_{1}^{2} + {〈 \tilde{x}, {\tilde{Γ}}^{- 1} \tilde{x} 〉}^{2})}^{1 / 2}, x = (x_{1}, \tilde{x}) \in R^{ν}

(38)

be a positive function on

R^{ν}

satisfying the homogeneity property:

ρ (λ x_{1}, λ^{1 / 2} \tilde{x}) = λ ρ (x),

\forall λ > 0

. As in Example 2 fractional coefficients for

p (u)

in (36) write as

\begin{matrix} τ (- u; d) & = & ψ_{u_{1}} (d) p_{u_{1}} (- u) I (u_{1} \geq 0), u = (u_{1}, \tilde{u}) \in Z^{ν} . \end{matrix}

(39)

Proposition 3.

Let satisfy (37) and

θ \in (0, 1)

. Then

τ (- u; d) = \frac{u_{1}^{- d - (ν + 1) / 2}}{Γ (- d) {(2 π θ)}^{(ν - 1) / 2} \sqrt{\det \tilde{Γ}}} exp \{- \frac{〈 \tilde{u}, {\tilde{Γ}}^{- 1} \tilde{u} 〉}{2 θ u_{1}}\} (1 + o (1))

(40)

u_{1} \to \infty

and

| \tilde{u} | \to \infty, | \tilde{u} | = o (u_{1}^{2 / 3}) .

We also have that

τ (- u; d) = ρ {(u)}^{- d - (ν + 1) / 2} (L_{0} (\frac{u_{1}}{ρ (u)}) + o (1)), | u | \to \infty,

(41)

where

L_{0} (z), z \in [- 1, 1]

is a continuous function on

[- 1, 1]

given by

L_{0} (z) : = \frac{z^{- d - (ν + 1) / 2}}{Γ (- d) {(2 π θ)}^{(ν - 1) / 2} \sqrt{\det \tilde{Γ}}} exp \{- (1 / 2 θ) \sqrt{{(1 / z)}^{2} - 1}\}

(42)

for

z \in (0, 1]

, and equal 0 for

z \in [- 1, 0]

Proof.

Consider the following j-step probabilities of a random walk on

Z^{ν - 1}

starting at

0

q_{j} (\tilde{u}) : = p_{j} (u)

, where

u = (- j, \tilde{u})

for

\tilde{u} \in Z^{ν - 1}

j = 0, 1, \dots

. Let us estimate them by

{\bar{q}}_{j} (\tilde{u}) : = \frac{1}{{(2 π j)}^{(ν - 1) / 2} \sqrt{\det Γ}} e^{- 〈 \tilde{u}, Γ^{- 1} \tilde{u} 〉 / 2 j}

, where

Γ

is the covariance matrix of the 1-step distribution

q_{1} (\tilde{u})

u \in Z^{ν - 1}

. Note

Γ = θ \tilde{Γ}

. By Lemma 2,

| q_{j} (\tilde{u}) - {\bar{q}}_{j} (\tilde{u}) | \leq C {\bar{q}}_{j} (\tilde{u}) (\frac{1}{j^{1 / 2}} + \frac{| \tilde{u} |^{3}}{j^{2}}), \forall | \tilde{u} | < j^{2}, \tilde{u} \in Z^{ν - 1} .

(43)

Relation (40) follows directly from (39), (43) and (3). Relation (41) writes as

ρ {(u)}^{d + (ν + 1) / 2} τ (- u; d) - L_{0} (\frac{u_{1}}{ρ (u)}) \to 0, | u | \to \infty .

(44)

The asymptotics in (44) is immediate from (40) for

| u |

tending to ∞ as in (40). The general case of (44) also follows from (40) using the continuity of

L_{0}

. For

ν = 2

the details can be found in [22]. □

Remark 2.

The approximation in (40) compares with the kernel

\begin{matrix} h_{c, - d} (t) & = & c_{1} t_{1}^{- d - \frac{1 + ν}{2}} exp {- c t_{1} - \frac{| \tilde{t} |^{2}}{4 t_{1}}} I (t_{1} > 0), t = (t_{1}, \tilde{t}) \in R^{ν} \end{matrix}

(45)

of the fractional heat operator

{(c + \partial_{1} - \tilde{Δ})}^{- d}, \partial_{1} - \tilde{Δ} : = \partial / \partial t_{1} - \sum_{i = 2}^{ν} \partial^{2} / \partial t_{i}^{2}

for all

c > 0, d < 0

and some

c_{1} \in R

. For

ν = 2

,24] has recently derived the analytic form in (45) of the kernel from the absolute square of its Fourier transform:

\begin{matrix} | {\hat{h}}_{c, - d} {(z) |}^{2} & = & | \int_{R^{ν}} e^{ß 〈 z, t 〉} h_{c, - d} (t) \underset{\cdot}{t} |^{2} = c_{1}^{2} {(4 π)}^{ν - 1} Γ {(- d)}^{2} {(z_{1}^{2} + (c + | \tilde{z} {|^{2})}^{2})}^{d}, z = (z_{1}, \tilde{z}) \in R^{ν}, \end{matrix}

(46)

which is implicit definition of this kernel in [16]. Similarly to derivations in [24], for

ν \geq 2

, table of integrals [15] [3.944.5-6] gives

\begin{matrix} {\hat{h}}_{c, - d} (z) & = & c_{1} \int_{0}^{\infty} e^{ß z_{1} t_{1} - c t_{1}} t_{1}^{d - \frac{1 + ν}{2}} \underset{\cdot}{t_{1}} \int_{R^{ν - 1}} exp {ß 〈 \tilde{z}, \tilde{t} 〉 - \frac{| \tilde{t} |^{2}}{4 t_{1}}} \underset{\cdot}{t} \\ = & c_{1} {(4 π)}^{\frac{ν - 1}{2}} \int_{0}^{\infty} e^{ß z_{1} t_{1} - t_{1} (c + | \tilde{z} |^{2})} t_{1}^{- d - 1} \underset{\cdot}{t} \\ = & c_{1} {(4 π)}^{\frac{ν - 1}{2}} Γ (- d) {(z_{1}^{2} + (c + | \tilde{z} {|^{2})}^{2})}^{\frac{d}{2}} exp {- ß d arctan (\frac{z_{1}}{c + | \tilde{z} |^{2}}}}, \end{matrix}

yielding (46).

Finally, the tempered fractional coefficients in (20) are summable:

\sum_{u \in Z^{ν}} | τ_{r} (u; d) | \leq \sum_{j = 0}^{\infty} r^{j} | ψ_{j} (d) | \leq 2 {(1 - r)}^{- | d |} < \infty

for any

d \in (- 1, 1), r \in (0, 1)

and any random walk

{S_{j}}

. Assuming the existence of exponential moment

E e^{κ | S_{1} |} < \infty

for some

κ > 0

, (20) decay exponentially

| τ_{r} (u; d) | \leq C e^{- c | u |}, u \in Z^{ν},

(47)

for some

C, c > 0

. Indeed, Markov’s inequality gives

r^{j} | ψ_{j} (d) | p_{j} (u) \leq P (| S_{j} | \geq | u |) \leq e^{- κ | u |} E e^{κ | S_{j} |} \leq e^{- κ | u |} {(E e^{κ | S_{1} |})}^{j} \leq e^{- (κ / 2) | u |}

for any

0 \leq j < c | u |

and large enough

| u |

. Moreover,

\sum_{j \geq c | u |} r^{j} | ψ_{j} (d) | p_{j} (u) \leq

\sum_{j \geq c | u |} r^{j} = r^{c | u |} / (1 - r)

, proving (47).

3. Fractionally Integrated Random Fields on $Z^{ν}$

Let

{ε (t); t \in Z^{ν}}

be a white noise, in other words, a sequence of r.v.s with

E ε (t) = 0, E ε (t) ε (s) = I (t = s), t, s \in Z^{ν}

. Given a sequence

a \in L^{2} (Z^{ν})

with the above noise we can associate a moving-average random field (RF)

X (t) = \sum_{s \in Z^{ν}} a (u) ε (t - u), t \in Z^{ν}

(48)

with zero mean and covariance

Cov (X (t), X (s)) = \sum_{u \in Z^{ν}} a (u) a (t - s + u)

, which depends on

t - s

alone and characterizes the dependence between values of X at distinct points

t \neq s

A moving-average RF X in (48) will be said to be

long-range dependent (LRD) if $\sum_{u \in Z^{ν}} | a (u) | = \infty$ ;
short-range dependent (SRD) if $\sum_{u \in Z^{ν}} | a (u) | < \infty, \sum_{u \in Z^{ν}} a (u) \neq 0$ ;
negatively dependent (ND) if $\sum_{u \in Z^{ν}} | a (u) | < \infty, \sum_{u \in Z^{ν}} a (u) = 0$ .

The above classification is important in limit theorems and applications of random fields. It is not unanimous; several related but not equivalent classifications of dependence for stochastic processes can be found in [14,18,25,30] and other works.

Many RF models with discrete argument are defined through linear difference equations involving white noise [13]. In this paper, we deal with fractionally integrated RFs X solving fractional equations on

Z^{ν}

\begin{matrix} {(I - T)}^{d} X (t) & = & \sum_{s \in Z^{ν}} τ (s; d) X (t + s) = ε (t), \end{matrix}

(49)

\begin{matrix} {(I - r T)}^{d} X (t) & = & \sum_{s \in Z^{ν}} τ_{r} (s; d) X (t + s) = ε (t), 0 < r < 1, t \in Z^{ν}, \end{matrix}

(50)

whose solutions are obtained by inverting these operator; see below.

Definition 1.

Let

d \in (- 1, 1)

and

τ (u; \pm d)

in (9) be well-defined. By stationary solution of Equation (49) (respectively, (50)) we mean a stationary RF X such that for each

t \in Z^{ν}

the series in (49) converges in mean square and (49) holds (respectively, the series in (50) converges in mean square and (50) holds).

Corollary 1.

(i) Let

- 1 < d < 1

. Then

X (t) = {(I - T)}^{- d} ε (t) = \sum_{u \in Z^{ν}} τ (u; - d) ε (t + u), t \in Z^{ν}

(51)

is a stationary solution of Equation (49) if condition (21) holds (for

0 < d < 1

, (21) is also necessary for the existence of the above X).

(ii) Let

0 < d < 1

and (21) hold. Then X in (51) is LRD. Moreover, it has nonnegative covariance function

Cov (X (0), X (t)) \geq 0

and

\sum_{t \in Z^{ν}} Cov (X (0), X (t)) = \infty

(iii) Let

- 1 < d < 0

and (21) hold. Then X in (51) is ND; moreover

\sum_{t \in Z^{ν}} Cov (X (0), X (t)) = 0

(iv) Let

- 1 < d < 1, 0 < r < 1

. Then

X (t) = {(I - r T)}^{- d} ε (t) = \sum_{u \in Z^{ν}} τ_{r} (u; - d) ε (t + u), t \in Z^{ν}

(52)

is a stationary solution of equation (50). Moreover, X in (52) is SRD and

\sum_{t \in Z^{ν}} | Cov (X (0), X (t)) | < \infty, \sum_{t \in Z^{ν}} Cov (X (0), X (t)) = {(1 - r)}^{- 2 d} > 0

Proof. (i) Let

0 < d < 1

. X in (51) is well-defined if and only if (22) holds, which is therefore a necessary condition. Let us show that X in (51) is a stationary solution of (49). We use the spectral representation of white noise

\begin{matrix} ε (t) = \int_{Π^{ν}} e^{i 〈 t, x 〉} Z (\underset{\cdot}{x}), t \in Z^{ν}, \end{matrix}

(53)

where

Z (x), x \in Π^{ν}

is a random complex-valued spectral measure with zero mean and variance

{E | Z (x) |}^{2} = \underset{\cdot}{x} / {(2 π)}^{ν}

. Then

X (t)

writes as

\begin{matrix} X (t) & = & \int_{Π^{ν}} e^{ß 〈 t, x 〉} \hat{τ} (x - d) Z (\underset{\cdot}{x}) = \int_{Π^{ν}} e^{ß 〈 t, x 〉} \frac{Z (\underset{\cdot}{x})}{{(1 - \hat{p} (x))}^{d}} \end{matrix}

(54)

see (23). Then

{(I - T)}^{d} X (t) = \int_{Π^{ν}} e^{ß 〈 t, x 〉} \sum_{s \in Z^{ν}} τ (s; d) e^{ß 〈 s, x 〉} {(1 - \hat{p} (x))}^{- d} Z (\underset{\cdot}{x}) = ε (t)

follows by (24) and absolute summability

\sum_{s \in Z^{ν}} | τ (s; d) | < \infty

, see (14), (18).

Next, let

- 1 < d < 0

. Then X in (51) is well-defined and writes as (54) due to

\sum_{s \in Z^{ν}} | τ (s; - d) | < \infty

. We need to show that the series in (49) converges in mean square towards

ε (t)

if and only if (21) or (22) hold. The latter convergence writes as

\begin{matrix} {lim}_{M \to \infty} E {| s_{M} - ε (t) |}^{2} = 0, where s_{M} : = \sum_{| s | \leq M} τ (s; d) X (t + s) . \end{matrix}

From (54),

\begin{matrix} E | s_{M} {- ε (t) |}^{2} & = & {(2 π)}^{- ν} \int_{Π^{ν}} | \sum_{| s | \leq M} e^{ß 〈 s, x 〉} τ (s; d) - {(1 - \hat{p} (x))}^{d} |^{2} {| 1 - \hat{p} (x) |}^{2 | d |} \underset{\cdot}{x} \\ \leq & C \int_{Π^{ν}} | \sum_{| s | \leq M} e^{ß 〈 s, x 〉} τ (s; d) - {(1 - \hat{p} (x))}^{d} |^{2} \underset{\cdot}{x} \\ = & C \int_{Π^{ν}} | \sum_{| s | > M} e^{ß 〈 s, x 〉} τ (s; d) |^{2} \underset{\cdot}{x} \\ = & C \sum_{| s | > M} τ {(s; - | d |)}^{2} \to 0 (M \to \infty) \end{matrix}

in view of (22). This proves part (i).

(ii) From (12), (9) we see that

τ (s; - d) \geq 0

are nonnegative and

\sum_{s \in Z^{ν}} τ (s; - d)

= \sum_{j = 0}^{\infty} ψ_{j} (- d)

= \infty

. Thus,

Cov (X (0), X (t)) = \sum_{s \in Z^{ν}} τ (s; - d) τ (t + s; - d) \geq 0

and

\sum_{t \in Z^{ν}} Cov (X (0), X (t)) = \infty

(iii) As in the proof of (i) we get that

\sum_{s \in Z^{ν}} | τ (s; - d) | \leq 1 + \sum_{j = 1}^{\infty} \sum_{s \in Z^{ν}} | ψ_{j} (- d) | p_{j} (s) = 1 + \sum_{j = 1}^{\infty} | ψ_{j} (- d) | = 2

, see (12), and

\sum_{s \in Z^{ν}} τ (s; - d) = 0

, implying

\sum_{t \in Z^{ν}} Cov (X (0), X (t))

= \sum_{t, s \in Z^{ν}} τ (s; - d) τ (t + s; - d) = 0

(iv) Using

\sum_{u \in Z^{ν}} | τ_{r} (u; d) | < \infty,

\sum_{u \in Z^{ν}} τ_{r} (u; d) = \sum_{j = 0}^{\infty} r^{j} ψ_{j} (d) = {(1 - r)}^{d}

the proof is similar as above. Corollary 1 is proved. □

ARFIMA(0,d,0) Equation (4) is autoregressive since the best linear predictor (or conditional expectation in the Gaussian case) of

X (t)

given the `past’

X (s), s < t

is a linear combination

\sum_{j = 1}^{\infty} ψ_{j} (d) X (t - j)

of the `past’ observations, due to the fact that

Cov (X (s), ε (t)) = 0 (s < t)

. For spatial equations as in (49) or (), an analogous property given the `past’

X (s), s \neq t

does not hold since

Cov (X (s), ε (t)) \neq 0 (s \neq t)

as a rule. This issue is important in spatial statistics and has been discussed in the literature, see [3,4] and the references therein, distinguishing between `simultaneous’ and `conditional autoregressive schemes’. The recent work [12] discusses some conditional autoregressive models with LRD property.

Definition 2.

Let X be an RF with

E X {(t)}^{2} < \infty

for each

t \in Z^{ν}

. We say that X has:

(i) a simultaneous autoregressive representation with coefficients

b (s), s \in Z_{0}^{ν}

if for each

t \in Z^{ν}

\begin{matrix} X (t) & = & \sum_{s \in Z_{0}^{ν}} b (s) X (t - s) + ξ (t), \end{matrix}

(55)

where the series converges in mean square and the r.v.s

ξ (t), t \in Z^{ν}

satisfy

Cov (ξ (t), ξ (s)) = 0 (\forall s \neq t)

(ii) a conditional autoregressive representation with coefficients

c (s), s \in Z_{0}^{ν}

if for each

t \in Z^{ν}

\begin{matrix} X (t) & = & \sum_{s \in Z_{0}^{ν}} c (s) X (t - s) + η (t), \end{matrix}

(56)

where the series converges in mean square and the r.v.s

η (t), t \in Z^{ν}

satisfy

Cov (η (t), X (s)) = 0 (\forall s \neq t)

Corollary 2.

(i) Let

d \in (- 1, 1)

and X be a fractionally integrated RF in (51) and (21) holds. Then X has a simultaneous autoregressive representation with coefficients

b (s) = - τ (- s; d) / τ (0; d), s \in Z_{0}^{ν}

and

ξ (t) = ε (t) / τ (0; d), s \in Z^{ν}

;

(ii) Let

d \in (0, 1)

, X be a fractionally integrated RF in (51) and (21) holds. Then X has a conditional autoregressive representation with coefficients

c (s) = - γ^{*} (s) / γ^{*} (0), s \in Z_{0}^{ν}

and

η (t) = \int_{Π^{ν}} e^{ß 〈 t, x 〉} {(1 - \hat{p} (- x))}^{d} Z (\underset{\cdot}{x}) / γ^{*} (0),

where

\begin{matrix} γ^{*} (s) & : = & \frac{1}{{(2 π)}^{ν}} \int_{Π^{ν}} e^{- ß 〈 s, x 〉} {| 1 - \hat{p} (x) |}^{2 d} \underset{\cdot}{x} \end{matrix}

(57)

(iii) Let

d \in (- 1, 1), 0 < r < 1

and X be a (tempered) fractionally integrated RF in (52). Then X has a simultaneous autoregressive representation with

b (s) = - τ_{r} (- s; d) / τ_{r} (0; d), ξ (t) = ε (t) / τ_{r} (0; d)

and a conditional autoregressive representation with

c (s) = - γ_{r}^{*} (s) / γ_{r}^{*} (0),

η (t) = \int_{Π^{ν}} e^{ß 〈 t, x 〉} {(1 - r \hat{p} (- x))}^{d} Z (\underset{\cdot}{x}) / γ_{r}^{*} (0),

with the same

Z (\underset{\cdot}{x})

as in part (ii) and

\begin{matrix} γ_{r}^{*} (s) & : = & \frac{1}{{(2 π)}^{ν}} \int_{Π^{ν}} e^{- i 〈 s, x 〉} {| 1 - r \hat{p} (x) |}^{2 d} \underset{\cdot}{x} . \end{matrix}

(58)

Proof.

(i) is obvious from Corollary 1 and (49),

τ (0; d) \neq 0

(ii) By (21),

c (s)

and

η (t)

are well-defined,

η (t) \in R

and

E η {(t)}^{2} < \infty

. The orthogonality relation

E X (t) η (s) = 0 (t \neq s)

follows from spectral representations in (54), (53):

\begin{matrix} E X (t) η (s) & = & \frac{1}{{(2 π)}^{ν} γ^{*} (0)} \int_{Π^{ν}} e^{ß 〈 t - s, x 〉} \frac{{(\bar{1 - \hat{p} (- x)})}^{d}}{{(1 - \hat{p} (x))}^{d}} \underset{\cdot}{x} \\ = & \frac{1}{{(2 π)}^{ν} γ^{*} (0)} \int_{Π^{ν}} e^{ß 〈 t - s, x 〉} \underset{\cdot}{x} = 0 (t \neq s) . \end{matrix}

It remains to show (56), including the convergence of the series. In view of the definition of

c (s)

this amounts to showing

\sum_{s \in Z^{ν}} X (t - s) γ^{*} (s) = γ^{*} (0) η (t)

or, in spectral terms, to the convergence of the Fourier series

\frac{1}{{(1 - \hat{p} (x))}^{d}} \sum_{s \in Z^{ν}} e^{- ß 〈 x, s 〉} γ^{*} (s) = {(1 - \hat{p} (- x))}^{d} = \frac{| 1 - \hat{p} {(- x) |}^{2 d}}{{(1 - \hat{p} (x))}^{d}}

(59)

L^{2} (Π^{ν})

. Note

γ^{*} (s) = Cov (X^{*} (0), X^{*} (s))

, where the RF

X^{*} (t) : = {(1 - T)}^{d} ε (t)

t \in Z^{ν}

, results from application of the inverse operator. Since

X^{*}

has negative dependence, see (54) and the proof of Corollary 1 (iii), the covariances

γ^{*} (s), s \in Z^{ν}

are absolutely summable. Therefore, the Fourier series on the l.h.s. of (59) converges uniformly in

x \in Π^{ν}

| 1 - \hat{p} {(- x) |}^{2 d}

, proving (59).

(iii) The proof is analogous (and simpler) as (i)-(ii), using

\sum_{u \in Z^{ν}} | τ_{r} (u; d) | < \infty

. □

Example 1.

Fractional Laplacian. The (lattice) Laplace operator on

Z^{ν}

is defined as

\begin{matrix} [Δ] g (t) & : = & \frac{1}{2 ν} \sum_{j = 1}^{ν} (g (t + e_{j}) + g (t - e_{j}) - 2 g (t)), t \in Z^{ν} \end{matrix}

(60)

so that

[Δ] = T - I

, where

T g (t) = \frac{1}{2 ν} \sum_{j = 1}^{ν} (g (t + e_{j}) + g (t - e_{j}))

is the transition operator of the simple random walk

{S_{j}; j = 0, 1, \dots}

Z^{ν}

with equal one-step transition probabilities

1 / 2 ν

to the nearest-neighbors

t \to t \pm e_{j}, j = 1, \dots, ν

. For

- 1 < d < 1

, the fractional Laplace RF can be defined as a stationary solution of the difference equation

{(- [Δ])}^{d} X (t) = ε (t), t \in Z^{ν}

(61)

with weak white noise on the r.h.s., written as a moving-average RF:

X (t) = {(- [Δ])}^{- d} ε (t) = \sum_{u \in Z^{ν}} τ (u; - d) ε (t + u) .

(62)

We find that

\hat{p} (x) = (1 / ν) \sum_{j = 1}^{ν} cos (x_{j})

x = (x_{1}, \dots, x_{ν}) \in Π^{ν}

and

\begin{matrix} 1 - \hat{p} (x) = \frac{1}{ν} \sum_{j = 1}^{ν} (1 - cos (x_{j})) \geq C {| x |}^{2} \end{matrix}

for some

C > 0

and

1 - \hat{p} (x) \sim {(1 / 2 ν) | x |}^{2} (| x | \to 0)

. Hence, condition (21) for (61) translates to

\begin{matrix} \int_{Π^{ν}} \frac{\underset{\cdot}{x}}{| 1 - \hat{p} {(x) |}^{2 | d |}} < \infty ⟺ | d | < \frac{ν}{4} . \end{matrix}

(63)

In particular, a stationary solution of the equation (61) on

ν \geq 4

exists for all

- 1 < d < 1

. Finally, recall that (21) is equivalent to the condition (22). We could have verified the latter by using Corollary 2, which gives the asymptotics of coefficients

τ (u; - d)

in (62).

Example 2.

Fractional heat operator. For a parameter

0 < θ < 1

, we can extend the definition of the (lattice) heat operator on

Z^{ν}

from

ν = 2

in [22] to

ν \geq 2

as follows:

\begin{matrix} Δ_{1, 2} g (t) & : = & (1 - θ) (g (t) - g (t - e_{1})) \\ - & \frac{θ}{2 (ν - 1)} \sum_{j = 2}^{ν} (g (t - e_{1} + e_{j}) + g (t - e_{1} - e_{j}) - 2 g (t)) . \end{matrix}

(64)

Thus,

Δ_{1, 2} = I - T

corresponds to the random walk on

Z^{ν}

with 1-step distribution

p (- e_{1}) = 1 - θ, p (- e_{1} \pm e_{j}) = \frac{θ}{2 (ν - 1)}, j = 2, \dots, ν

. We find that

\begin{matrix} | 1 - \hat{p} {(x) |}^{2} & = & {(cos (x_{1}) - 1 + \frac{θ}{ν - 1} \sum_{j = 2}^{ν} (1 - cos (x_{j})))}^{2} + {sin}^{2} (x_{1}), x = (x_{1}, \dots, x_{ν}) \in Π^{ν} . \end{matrix}

By Taylor expansion,

\begin{matrix} | 1 - \hat{p} {(x) |}^{2} & \sim & {(\frac{θ}{2 (ν - 1)})}^{2} {| \tilde{x} |}^{4} + x_{1}^{2}, x \to 0, \tilde{x} : = (0, x_{2}, \dots, x_{ν}) . \end{matrix}

We also find that outside the origin

| 1 - \hat{p} {(x) |}^{2} \geq C

for some

C > 0

since

0 < θ < 1

. Therefore,

\begin{matrix} \int_{Π^{ν}} \frac{\underset{\cdot}{x}}{| 1 - \hat{p} {(x) |}^{2 | d |}} & \leq & C \int_{0}^{1} \int_{0}^{1} \frac{y^{ν - 2} \underset{\cdot}{x} \underset{\cdot}{y}}{{(x^{2} + y^{4})}^{| d |}} < \infty if | d | < \frac{ν + 1}{4} \end{matrix}

and

\int_{Π^{ν}} {| 1 - \hat{p} (x) |}^{- 2 | d |} \underset{\cdot}{x} = \infty

| d | \geq \frac{ν + 1}{4}

. The above result agrees with [22] for

ν = 2

0 < d < \frac{3}{4}

, and extends it to arbitrary

ν \geq 2

- 1 < d < 1

Example 3.

Fractionally integrated time series models (case

ν = 1

). As noted above, ARFIMA

(0, d, 0)

process is a particular case of (51) corresponding to backward shift

T g (t) : = g (t - 1)

or deterministic random walk

t \to t - 1

. Another fractionally integrated time series model is given in Example 1 and corresponds to the symmetric nearest-neighbor random walk on

Z

with probabilities 1/2. It is of interest to compare these two processes and their properties. Let

T_{1} g (t) : = g (t - 1), T_{2} g (t) : = (1 / 2) (g (t + 1) + g (t - 1)), t \in Z

be the corresponding operators,

\begin{matrix} X_{1} (t) & : = & {(I - T_{1})}^{- d_{1}} ε (t) = \sum_{u = 0}^{\infty} ψ_{u} (- d_{1}) ε (t - u), \\ X_{2} (t) & : = & {(I - T_{2})}^{- d_{2}} ε (t) = \sum_{u \in Z} τ (u; - d_{2}) ε (t + u), t \in Z . \end{matrix}

For

| d_{1} | < 1 / 2

and

| d_{2} | < 1 / 4

, processes

X_{1}

and

X_{2}

are well-defined; moreover, they are stationary solutions of respective equations

{(I - T_{1})}^{d_{1}} X (t) = ε (t)

and

{(I - T_{2})}^{d_{2}} X (t) = ε (t)

. The spectral densities of

X_{1}

and

X_{2}

are given by

\begin{matrix} f_{1} (x) & = & \frac{1}{2 π | 1 - e^{- ß x} |^{2 d_{1}}} = \frac{1}{2 π \cdot 2^{d_{1}} {| 1 - cos (x) |}^{d_{1}}}, \\ f_{2} (x) & = & \frac{1}{2 π | 1 - (1 / 2) (e^{- ß x} + e^{ß x}) |^{2 d_{2}}} = \frac{1}{{2 π | 1 - cos (x) |}^{2 d_{2}}} \end{matrix}

We see that when

d_{1} = 2 d_{2}

processes

X_{1}

and

X_{2}

have the same 2nd order properties up to a multiplicative constant so that in the Gaussian case

X_{2}

is a noncausal representation of ARFIMA

(0, 2 d_{2}, 0)

4. Scaling Limits

As explained in the Introduction, isotropic scaling limits refer to the limits distribution of integrals

\begin{matrix} X_{λ} (ϕ) & : = & \int_{R^{ν}} X ([t]) ϕ (t / λ) \underset{\cdot}{t}, as λ \to \infty, \end{matrix}

(65)

where

X = {X (t); t \in Z^{ν}}

is a given stationary random field (RF), for each

ϕ : R^{ν} \to R

from a class of (test) functions

Φ

. We choose the latter class to be

Φ : = L^{1} (R^{ν}) \cap L^{\infty} (R^{ν}) .

(66)

In as follows, X is a linear or moving-average RF on

Z^{ν}

X (t) = \sum_{s \in Z^{ν}} a (t - s) ε (s), t \in Z^{ν},

(67)

where

{ε (t); t \in Z^{ν}}

are independent identically distributed (i.i.d.) r.v.s with

E ε (t) = 0, E ε {(t)}^{2} = 1

and

a \in L^{2} (Z^{ν})

are deterministic coefficients. Obviously, stationary solution (51) of Equation (49) satisfying Corollary 1 is a particular case of linear RF with

a (t) = τ (- t; - d)

. Our limits results assume an `isotropic’ behavior of

a (t)

| t | \to \infty

detailed in as follows. Let

C (S_{ν - 1})

denote the class of all continuous functions on

S_{ν - 1} = {t \in R^{ν} : | t | = 1}

Assumption (A) $(d)$ Let

{a (t); t \in Z^{ν}}

be a sequence of real numbers satisfying the following properties.

(i) Let

0 < d < ν / 4

. Then

a (t) = \frac{1}{{| t |}^{ν - 2 d}} (ℓ (\frac{t}{| t |}) + o (1)), | t | \to \infty,

(68)

where

ℓ (\cdot) \in C (S_{ν - 1})

is not identically zero.

(ii) Let

- ν / 4 < d < 0

. Then

a (t)

satisfy (68) with the same

ℓ (t)

and, moreover,

\sum_{t \in Z^{ν}} a (t) = 0

(iii) Let

d = 0

. Then

\sum_{t \in Z^{ν}} | a (t) | < \infty

and

\sum_{t \in Z^{ν}} a (t) \neq 0

The class of RFs in (67) with coefficients satisfying Assumption (A)

(d)

is related but not limited to fractionally integrated RFs in (49)- (50). Note that the parameter d is no longer restricted to be in

(- 1, 1)

. By easy observation, Assumption (A)

(d)

implies LRD, ND, and SRD properties of Section 3 in respective cases

d > 0, d < 0

, and

d = 0

. Following the terminology in time series [14], the parameter d in (68) may be called the memory parameter of the linear RF X in (67), except that for

ν = 1

the memory parameter usually is defined as

2 d \in (- 1 / 2, 1 / 2)

In particular, the covariance function

r (t) : = Cov (X (0), X (t))

of linear RF X in (67) writes as

\begin{matrix} r (t) & = & \sum_{u \in Z^{ν}} a (u) a (t + u), t \in Z^{ν} \end{matrix}

or the lattice convolution of

a (t)

with itself. We will use the notation

[a_{1} ★ a_{2}]

for lattice convolution and

(a_{1} ★ a_{2})

for continuous convolution, viz.,

\begin{matrix} [a_{1} ★ a_{2}] (t) & : = & \sum_{u \in Z^{ν}} a_{1} (u) a_{2} (t + u), t \in Z^{ν}, \\ (a_{1} ★ a_{2}) (t) & : = & \int_{R^{ν}} a_{1} (u) a_{2} (t + u) \underset{\cdot}{u}, t \in R^{ν} \end{matrix}

(69)

which is well defined for any

a_{i} \in L^{2} (Z^{ν}), i = 1, 2

(respectively, for any

a_{i} \in L^{2} (R^{ν})

i = 1, 2

Proposition 4.

Let

a_{i} \in L^{2} (Z^{ν})

satisfy Assumption (A)

(d)

with

0 < d < ν / 4

and some

ℓ_{i} \in C (S_{ν - 1}), i = 1, 2 .

Then

\begin{matrix} [a_{1} ★ a_{2}] (t) & = & {| t |}^{4 d - ν} (L_{12} (\frac{t}{| t |}) + o (1)), | t | \to \infty, \end{matrix}

(70)

where the (angular) function

L_{12} (\cdot) \in C (S_{ν - 1})

is given by

\begin{matrix} L_{12} (t) & : = & \int_{R^{ν}} \frac{ℓ_{1} (s / | s |) ℓ_{2} ((t - s) / | t - s |)}{{| s |}^{ν - 2 d} {| t - s |}^{ν - 2 d}} \underset{\cdot}{s}, t \in S_{ν - 1} . \end{matrix}

(71)

Proof.

The existence and continuity of

L_{12}

follow from finiteness of integrals

\int_{| s | < 1} {| s |}^{2 d - ν} \underset{\cdot}{s} < \infty

and

\int_{| s | > 1} {| s |}^{2 (2 d - ν)} \underset{\cdot}{s} < \infty

. For (70) it suffices to show that

\begin{matrix} {| t |}^{ν - 4 d} [a_{1} ★ a_{2}] (t) - L_{12} (t / | t |) & \to & 0, | t | \to \infty . \end{matrix}

(72)

Let

{| t |}_{+} : = | t | \lor 1

and

a_{i}^{0} (t) : = {| t |}_{+}^{2 d - ν} ℓ_{i} {(t / | t |}_{+}), a_{i}^{1} (t) : = a_{i} (t) - a_{i}^{0} (t) = {o (| t |}^{2 d - ν}), i = 1, 2

, see (68). Then

[a_{1} ★ a_{2}] (t) = \sum_{i, j = 0}^{1} [a_{1}^{i} ★ a_{2}^{j}] (t)

. Clearly, (72) follows from

\begin{matrix} {| t |}^{ν - 4 d} [a_{1}^{0} ★ a_{2}^{0}] (t) - L_{12} (t / | t |) & \to & 0, | t | \to \infty \end{matrix}

(73)

and

\begin{matrix} [a_{1}^{i} ★ a_{2}^{j}] (t) & = & o (| t |^{4 d - ν}), | t | \to \infty, (i, j) \neq (0, 0), i, j = 0, 1 . \end{matrix}

(74)

To prove (73) rewrite

[a_{1}^{0} ★ a_{2}^{0}] (t) = \int_{R^{ν}} a_{1}^{0} ([u]) a_{2}^{0} (t + [u]) \underset{\cdot}{u}

as integral and change the variable

u \to | t | u

in it. This leads to

{| t |}^{ν - 4 d} [a_{1}^{0} ★ a_{2}^{0}] (t) = {\tilde{L}}_{t} (t / | t |)

where

\begin{matrix} {\tilde{L}}_{t} (z) & : = & \int_{R^{ν}} a_{1, t} (\tilde{u}) a_{2, t} (z + \tilde{u}) \underset{\cdot}{u}, z \in S_{ν - 1} \end{matrix}

(75)

where

\begin{matrix} a_{i, t} (\tilde{u}) & : = & \frac{1}{{(| t |}^{- 1} \lor | \tilde{u} {|)}^{ν - 2 d}} ℓ_{i} (\frac{\tilde{u}}{{| t |}^{- 1} \lor | \tilde{u} |}), \tilde{u} : = \frac{[| t | u]}{| t |} \end{matrix}

Relation (73) follows once we prove the uniform convergence

{sup}_{z \in S_{ν - 1}} | {\tilde{L}}_{t} (z) - L_{12} (z) |

\to 0 (| t |

\to \infty)

. Since

S_{ν - 1}

is a compact set and

L_{12}

is continuous, the last relation is implied by the sequentional convergence

\begin{matrix} | {\tilde{L}}_{t} (z_{t}) - L_{12} (z) | & \to & 0 (| t | \to \infty) \end{matrix}

(76)

for any

z \in S_{ν - 1}

and any

{z_{t}}

convergent to

z

| z_{t} - z | \to 0 (| t | \to \infty)

. The proof of (76) uses the bound

\begin{matrix} | a_{i, t} (\tilde{u}) {| \leq C | u |}^{2 d - ν}, u \in R^{ν}, i = 1, 2, \end{matrix}

(77)

which follows from boundedness of

ℓ_{i}

and

| u | \leq | \tilde{u} | + | u - \tilde{u} |

with

| u - \tilde{u} | \leq ν^{1 / 2} / | t |

hence

| u | \leq ν^{1 / 2} (| \tilde{u} {| + | t |}^{- 1}) \leq 2 ν^{1 / 2} (| \tilde{u} {| \lor | t |}^{- 1})

. Note

a_{1, t} (\tilde{u}) a_{2, t} (z + \tilde{u}) \to a_{1}^{0} (u) a_{2}^{0} (z + u) (| t | \to \infty)

for any

u \neq 0, z

and

| a_{1, t} (\tilde{u}) a_{2, t} (z + \tilde{u}) {| \leq C | u |}^{2 d - ν} {| z + u |}^{2 d - ν}

according to (77). Since

h (u) : = {C | u |}^{2 d - ν} {| z + u |}^{2 d - ν}

does not depend on

t

and

\int_{R^{ν}} h (u) \underset{\cdot}{u} < \infty

, Pratt’s lemma [26] applies to the integral in (75) resulting in (76) and (73). The proof of (74) is similar and simpler and is omitted. □

The question about the asymptotics of the variance of (65) arises assuming the power-law asymptotics of the covariance admitting a power-law behavior at large lags which is tackled in the following proposition.

Proof.

(i) For any

β > 0, ϕ_{i} \in Φ, i = 1, 2

λ \to \infty

\begin{matrix} \int_{R^{2 ν}} | ϕ_{1} (t_{1} / λ) ϕ_{2} (t_{2} / λ) | (1 \land | t_{1} - t_{2} |^{- β}) {\underset{\cdot}{t}}_{1} {\underset{\cdot}{t}}_{2} & = & \{\begin{matrix} O (λ^{ν}), & β > ν, \\ O (λ^{2 ν - β}), & β < ν, \\ O (λ^{ν} log λ), & β = ν . \end{matrix} \end{matrix}

(78)

(ii) Let

r (t), t \in Z^{ν}

satisfy

\begin{matrix} r (t) & = & {| t |}^{4 d - ν} (L (\frac{t}{| t |}) + o (1)), | t | \to \infty, \end{matrix}

(79)

where

0 < d < ν / 4

and

L \in C (S_{ν - 1})

. Then for any

ϕ_{i} \in Φ, i = 1, 2

\begin{matrix} lim_{λ \to \infty} λ^{- ν - 4 d} \int_{R^{2 ν}} ϕ_{1} (t_{1} / λ) ϕ_{2} (t_{2} / λ) r ([t_{1}] - [t_{2}]) {\underset{\cdot}{t}}_{1} {\underset{\cdot}{t}}_{2} & = & c (ϕ_{1}, ϕ_{2}), \end{matrix}

(80)

where

c (ϕ_{1}, ϕ_{2}) : = \int_{R^{2 ν}} ϕ_{1} (t_{1}) ϕ_{2} (t_{2}) L (\frac{t_{1} - t_{2}}{| t_{1} - t_{2} |}) \frac{{\underset{\cdot}{t}}_{1} {\underset{\cdot}{t}}_{2}}{| t_{1} - t_{2} |^{ν - 4 d}} .

(81)

(iii) Let

r \in L^{1} (Z^{ν})

. Then for any

ϕ_{i} \in Φ, i = 1, 2

\begin{matrix} lim_{λ \to \infty} λ^{- ν} \int_{R^{2 ν}} ϕ_{1} (t_{1} / λ) ϕ_{2} (t_{2} / λ) r ([t_{1}] - [t_{2}]) {\underset{\cdot}{t}}_{1} {\underset{\cdot}{t}}_{2} & = & \int_{R^{ν}} ϕ_{1} (t) ϕ_{2} (t) \underset{\cdot}{t} \times \sum_{s \in Z^{ν}} r (s) . \end{matrix}

(82)

Proof.

(i) Write

I_{λ, β}

for the l.h.s. of (78). Let first

β > ν

. Then

I_{λ, β} \leq C \int_{R^{ν}} | ϕ_{1} (t_{1} / λ) | {\underset{\cdot}{t}}_{1} \times

\int_{R^{ν}} 1 \land | t_{2} - t_{1} |^{- β} {\underset{\cdot}{t}}_{2} \leq C \int_{R^{ν}} | ϕ_{1} (t_{1} / λ) | {\underset{\cdot}{t}}_{1} = C λ^{ν} \int_{R^{ν}} | ϕ_{1} (t) | \underset{\cdot}{t} = O (λ^{ν})

\int_{R^{ν}} 1 \land {| t |}^{- β} \underset{\cdot}{t} < \infty

. Next, let

β < ν

then

I_{λ, β} \leq λ^{2 ν - β} J_{β}

, where

J_{β} : = \int_{R^{2 ν}} | ϕ_{1} (t_{1}) ϕ_{2} (t_{2}) | | t_{1} - t_{2} |^{- β} {\underset{\cdot}{t}}_{1} {\underset{\cdot}{t}}_{2} < \infty

follows by

J_{β} \leq C \int_{R^{ν}} | ϕ_{1} (t_{1}) | {\underset{\cdot}{t}}_{1} \int_{| t_{2} - t_{1} | \leq 1} {| t_{2} - t_{1} |}^{- β} {\underset{\cdot}{t}}_{2}

\int_{R^{2 ν}} | ϕ_{1} (t_{1}) ϕ_{2} (t_{2}) | \underset{\cdot}{t} {\underset{\cdot}{t}}_{2} < \infty

. Finally, for

β = ν

we have

I_{λ, ν} = λ^{ν} J_{λ, ν}

where

J_{λ, ν} : = \int_{R^{2 ν}} | ϕ_{1} (t_{1}) ϕ_{2} (t_{2}) | (λ^{- 1} \lor | t_{1} - t_{2} {|)}^{- ν} {\underset{\cdot}{t}}_{1} {\underset{\cdot}{t}}_{2} = O (log λ)

follows similarly.

(ii) The convergence of the integral in (81) follows from that of

J_{β}

in part (i), with

β = ν - 4 d .

Let

c_{λ} (ϕ_{1}, ϕ_{2})

denote the integral on the l.h.s. of (80). By change of variables,

\begin{matrix} \frac{c_{λ} (ϕ_{1}, ϕ_{2})}{λ^{ν + 4 d}} & = & \int_{R^{2 ν}} \frac{ϕ (t_{1}) ϕ (t_{2})}{| t_{1} - t_{2} |^{ν - 4 d}} {\tilde{L}}_{λ} (t_{1}, t_{2}) {\underset{\cdot}{t}}_{1} {\underset{\cdot}{t}}_{2}, \end{matrix}

where

{\tilde{L}}_{λ} (t_{1}, t_{2}) \to L (\frac{t_{1} - t_{2}}{| t_{1} - t_{2} |}) (λ \to \infty)

for any

t_{1} \neq t_{2}

. Using Pratt’s lemma [26], it suffices to prove (80) for

L \equiv 1

. In the latter case and with

{\tilde{t}}_{i} : = [λ t_{i}] / λ, i = 1, 2

we see that

| {\tilde{L}}_{λ} (t_{1}, t_{2}) | \leq C {(| t_{1} - t_{2} | / (| {\tilde{t}}_{1} - {\tilde{t}}_{2} | \lor (1 / λ)))}^{ν - 4 d} \leq C

as in the proof of Proposition 4. Thus, (80) follows from the DCT.

(iii) Let

c_{λ} (ϕ_{1}, ϕ_{2})

be the same as in the proof of (ii). For a large

K > 0

, write

c_{λ} (ϕ_{1}, ϕ_{2}) = \sum_{i = 1}^{3} c_{i, λ},

where

c_{1, λ} : = \int_{| t_{1} - t_{2} | > K} ϕ_{1} (t_{1} / λ) ϕ_{2} (t_{2} / λ) r ([t_{1}] - [t_{2}]) {\underset{\cdot}{t}}_{1} {\underset{\cdot}{t}}_{2},

c_{2, λ} : = \int_{| t_{1} - t_{2} | \leq K} ϕ_{1} (t_{1} / λ)

\times ϕ_{2} (t_{1} / λ) r ([t_{1}] - [t_{2}]) {\underset{\cdot}{t}}_{1} {\underset{\cdot}{t}}_{2},

and

c_{3, λ} : = \int_{| t_{1} - t_{2} | \leq K} ϕ_{1} (t_{1} / λ) (ϕ_{2} (t_{2} / λ) - ϕ_{2} (t_{1} / λ)) r ([t_{1}] - [t_{2}]) {\underset{\cdot}{t}}_{1} {\underset{\cdot}{t}}_{2}

. Here,

λ^{- ν} | c_{1, K} | \leq C λ^{- ν} \int_{R^{ν}} | ϕ_{1} (t / λ) | \underset{\cdot}{t} \sum_{| s | > K} | r (s) | \leq C \sum_{| s | > K} | r (s) |

can be made arbitrary small uniformly in

λ \geq 1

by choosing K large enough. Next,

\begin{matrix} λ^{- ν} | c_{3, λ} | & \leq & C \int_{R^{ν}} | ϕ_{1} (t) | \underset{\cdot}{t} \int_{| s | \leq K} | ϕ_{2} (t + \frac{s}{λ}) - ϕ_{2} (t) | \underset{\cdot}{s} . \end{matrix}

By boundedness of

ϕ_{2}

we see that the integral

\int_{| s | \leq K} | ϕ_{2} (t + \frac{s}{λ}) - ϕ_{2} (t) | \underset{\cdot}{s} \to 0 (λ \to \infty)

a.e. in

R^{ν}

, and is bounded in

t \in R^{ν}

. Then, since

ϕ_{1} \in L^{1} (R^{ν})

we conclude

{lim}_{λ \to \infty} λ^{- ν} | c_{3, λ} | = 0

by the DCT. Finally,

λ^{- ν} c_{2, λ} = \int_{R^{ν}} ϕ_{1} (t) ϕ_{2} (t) \underset{\cdot}{t} \int_{| s + [λ t] - λ t | \leq K} r (- [s]) \underset{\cdot}{s}

, and we can replace the last integral by the r.h.s. of (82) uniformly in

λ

provided K is large enough. □

Proposition 5 does not apply to ND covariances satisfying (79) with negative

d < 0

. This case is more delicate since it requires additional regularity conditions of test functions and the occurrence of `edge effects’. A detailed analysis of this issue in dimension

ν = 2

and for indicator (test) functions of rectangles in

R_{+}^{2}

can be found in [33]. Below we present a result in this direction and sufficient conditions on

d, ϕ_{i}, i = 1, 2

when the limits take a similar form to (80). Introduce a subclass of test functions:

\begin{matrix} Φ_{-} & : = & \{ϕ \in Φ : \int_{R^{ν}} {(\int_{R^{ν}} {| ϕ (t + s) - ϕ (s) |}^{2} \underset{\cdot}{s})}^{1 / 2} {| t |}^{2 d - ν} \underset{\cdot}{t} < \infty\} . \end{matrix}

(83)

Proposition 6.

Let

a \in L^{2} (Z^{ν})

satisfy Assumption (A)

(d)

with

- ν / 4 < d < 0

. Then for any

ϕ_{i} \in Φ_{-}, i = 1, 2

we have that

\begin{matrix} lim_{λ \to \infty} λ^{- ν - 4 d} \int_{R^{2 ν}} ϕ_{1} (t_{1} / λ) ϕ_{2} (t_{2} / λ) [a ★ a] ([t_{1}] - [t_{2}]) {\underset{\cdot}{t}}_{1} {\underset{\cdot}{t}}_{2} & = & c_{-} (ϕ_{1}, ϕ_{2}), \end{matrix}

(84)

where

\begin{matrix} c_{-} (ϕ_{1}, ϕ_{2}) : = \int_{R^{ν}} \prod_{i = 1}^{2} (\int_{R^{ν}} (ϕ_{i} (t + s) - ϕ_{i} (s)) {| t |}^{2 d - ν} ℓ (\frac{t}{| t |}) \underset{\cdot}{t}) \underset{\cdot}{s} . \end{matrix}

(85)

Proof.

The convergence of the integral on the r.h.s. of (85) follows from (83) and Minkowski’s integral inequality:

{\{\int_{R^{ν}} {(\int_{R^{ν}} {| ϕ (t + s) - ϕ (s) | | t |}^{2 d - ν} \underset{\cdot}{t})}^{2} \underset{\cdot}{s}\}}^{1 / 2} \leq \int_{R^{ν}} {∥ ϕ (t + \cdot) - ϕ (\cdot) ∥}_{L^{2} (R^{ν})}

\times {| t |}^{2 d - ν} \underset{\cdot}{t}

The proof of the convergence in (84) resembles that of (80). Write

c_{λ} (ϕ_{1}, ϕ_{2})

for the integral on the l.h.s. of (84). Using

\sum_{s \in Z^{ν}} a (s) = 0

we rewrite

\int_{R^{ν}} ϕ_{i} (t_{i} / λ) a ([t_{i}] - [s]) {\underset{\cdot}{t}}_{i} = \int_{R^{ν}} (ϕ_{i} ((t_{i} + s) / λ) - ϕ_{i} (s / λ)) a ([t_{i} + s] - [s]) {\underset{\cdot}{t}}_{i}

i = 1, 2, s \in R^{ν}

, and

\begin{matrix} \frac{c_{λ} (ϕ_{1}, ϕ_{2})}{λ^{ν + 4 d}} & = & \int_{R^{ν}} \underset{\cdot}{s} \prod_{i = 1}^{2} \int_{R^{ν}} (ϕ_{i} (t_{i} + s) - ϕ_{i} (s)) λ^{ν - 2 d} a ([λ (t_{i} + s)] - [λ s]) {\underset{\cdot}{t}}_{i}, \end{matrix}

where the inner integrals tend to those on the r.h.s. of (85) at each

s

such that

\int_{R^{ν}} | ϕ_{i} (t + s) - ϕ_{i} {(s) | | t |}^{2 d - ν} \underset{\cdot}{t} < \infty

i = 1, 2

. The remaining details are similar as in (80) and omitted. □

Remark 3.

The restriction

d > - ν / 4

in Proposition 6 is not necessary for (85). Indeed, if

ϕ \in Φ

satisfies the uniform Lipschitz condition

| ϕ (t) - ϕ (s) | < C (| t | < 1, s \in R^{ν})

then the integral in (83) converges for

0 > d > - ν / 2

implying

ϕ \in Φ_{-}

. On the other hand, for indicator functions

ϕ (t) = I (t \in A)

of a bounded Borel set

A \subset R^{ν}

with `regular’ boundary, we typically have

{∥ ϕ (t + \cdot) - ϕ (\cdot) ∥}_{L^{2} (R^{ν})} = {O (| t |}^{1 / 2})

leading to

d > - ν / 4

Relation (68) entails the existence of the scaling limit

lim_{λ \to \infty} λ^{ν - 2 d} a ([λ t]) = a_{\infty} (t) : = {| t |}^{2 d - ν} ℓ (\frac{t}{| t |}), λ \to \infty, \forall t \in R^{ν} ∖ {0} .

(86)

which is a continuous homogeneous function on

R^{ν}

: for any

λ > 0

we have that

a_{\infty} (λ t) = λ^{2 d - ν} a_{\infty} (t), t \in R^{ν} ∖ {0} .

(87)

With the limit function in (86) we associate a Gaussian RF:

W_{d} (ϕ) : = \{\begin{matrix} \int_{R^{ν}} (a_{\infty} ★ ϕ) (u) W (\underset{\cdot}{u}), & 0 < d < ν / 4, ϕ \in Φ \\ \int_{R^{ν}} {(a_{\infty} ★ ϕ)}_{reg} (u) W (\underset{\cdot}{u}), & - ν / 4 < d < 0, ϕ \in Φ_{-}, \\ \int_{R^{ν}} ϕ (u) W (\underset{\cdot}{u}), & d = 0, ϕ \in Φ, \end{matrix}

(88)

where

W (\underset{\cdot}{u})

is a real-valued Gaussian white noise with zero mean and variance

\underset{\cdot}{u}

(a_{\infty} ★ ϕ) (u) = \int_{R^{ν}} a_{\infty} (t) ϕ (t + u) \underset{\cdot}{t}

is the usual and

{(a_{\infty} ★ ϕ)}_{reg} (u) : = \int_{R^{ν}} a_{\infty} (t) (ϕ (t + u) - ϕ (u)) \underset{\cdot}{t}, u \in R^{ν}

(89)

the `regularized’ convolution. For indicator test function

ϕ (t) = I (t \in B)

of a Borel set

B \subset R^{ν}

(belonging to

Φ_{-}

) we see that the latter convolution equals

\begin{matrix} {(a_{\infty} ★ ϕ)}_{reg} (u) & = & \{\begin{matrix} \int_{B} a_{\infty} (t - u) \underset{\cdot}{t}, & u \notin B, \\ - \int_{R^{ν} ∖ B} a_{\infty} (t - u) \underset{\cdot}{t}, & u \in B . \end{matrix} \end{matrix}

(90)

The existence of stochastic integrals in (88) of follows from Propositions 5 and 6. Particularly, the variances

E W_{d}^{2} (ϕ) = c (ϕ, ϕ) (0 < d < ν / 4)

and

E W_{d}^{2} (ϕ) = c_{-} (ϕ, ϕ) (- ν / 4 < d < 0)

agree with (81) and (85).

Let

S (R^{ν})

be the Schwartz space of all infinitely differentiable rapidly decreasing functions

ϕ : R^{ν} \to R

. Following [9] we say that a generalized RF

Y = {Y (ϕ); ϕ \in S (R^{ν})}

is stationary if

Y (ϕ) \begin{matrix} \overset{d}{=} \end{matrix} Y (ϕ (\cdot + a)) (\forall ϕ \in S (R^{ν}), a \in R^{ν})

and H-self-similar (

H \in R

) if

Y (ϕ) \begin{matrix} \overset{d}{=} \end{matrix} λ^{H - ν} Y (ϕ (\cdot / λ)) (\forall ϕ \in S (R^{ν}), λ > 0)

. As noted in Remark 3,

S (R^{ν}) \subset Φ_{-} \subset Φ

, hence (88) are well-defined for any

ϕ \in S (R^{ν})

and represent stationary generalized RFs on

S (R^{ν})

. By scaling property in (87) and a change of variables we see that

W_{d} (ϕ) \begin{matrix} \overset{d}{=} \end{matrix} λ^{H (d) - ν} W_{d} (ϕ (\cdot / λ)) (\forall ϕ \in S (R^{ν}))

, hence RF

W_{d}

in (88) is

H (d)

-self-similar, with

H (d) : = \frac{ν - 4 d}{2} \in (0, ν), - ν / 4 < d < ν / 4 .

The RF in (88) appear as scaling limits in the following corollary.

Corollary 3.

Let X be a linear RF satisfying Assumption (A)

(d)

and

X_{λ} (ϕ)

be defined in (65). Then

\begin{matrix} λ^{- (ν + 4 d) / 2} X_{λ} (ϕ) \begin{matrix} \overset{d}{⟶} \end{matrix} \{\begin{matrix} W_{d} (ϕ), & 0 < d < ν / 4, \forall ϕ \in Φ, \\ W_{d} (ϕ), & - ν / 4 < d < 0, \forall ϕ \in Φ_{-}, \\ σ W_{0} (ϕ), & d = 0, \forall ϕ \in Φ, \end{matrix} \end{matrix}

where

σ^{2} : = {(\sum_{t \in Z^{ν}} a (t))}^{2}

Proof.

Since (65) writes as a linear form

X_{λ} (ϕ) = \sum_{u \in Z^{ν}} ε (u) \int_{R^{ν}} ϕ (t / λ) a ([t] - u) \underset{\cdot}{t}

in i.i.d. r.v.s, we can use the Lindeberg type condition, see also [14]. Accordingly, it suffices to show that

sup_{u \in Z^{ν}} | \int_{R^{ν}} ϕ (t / λ) a ([t] - u) \underset{\cdot}{t} | = o (\sqrt{Var (X_{λ} (ϕ))}), λ \to \infty

(91)

holds in each case

d > 0, d < 0, d = 0

of the corollary. The behavior of the last variance is detailed in Propositions 5 and 6 and it grows to infinity in each case of d. On the other hand, the l.h.s. of (91) does not exceed

{∥ ϕ ∥}_{L^{\infty} (R^{ν})} {∥ a ∥}_{L^{1} (Z^{ν})}

which is bounded in cases

d < 0

and

d = 0

. Finally, in case

d > 0

we see that the l.h.s. of (91) does not exceed

{∥ ϕ (\cdot / λ) ∥}_{L^{2} (R^{ν})} {∥ a ∥}_{L^{2} (Z^{ν})} = O (λ^{ν / 2})

and (91) holds since

d > 0

. □

Acknowledgments

We thank Maria Eulalia Vares for useful reference.

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Fractional Operators and Fractionally Integrated Random Fields on Z

Abstract

1. Introduction

2. Invertibility and Properties of Fractional Operators

Lemma 1.

3. Fractionally Integrated Random Fields on Z ν

4. Scaling Limits

Acknowledgments

References

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3. Fractionally Integrated Random Fields on $Z^{ν}$