Sample Path Properties of Generalized Random Sheets with Operator Scaling

https://doi.org/10.1007/s10959-020-01045-6

Sample Path Properties of Generalized Random Sheets with Operator Scaling

Ercan Sönmez¹

Received: 21 November 2018 / Revised: 16 August 2019 / Published online: 3 November 2020

© The Author(s) 2020

Abstract

We consider operator scalingα-stable random sheets, which were introduced in Hoff- mann (Operator scaling stable random sheets with application to binary mixtures.

Dissertation Universität Siegen, 2011). The idea behind such fields is to combine the properties of operator scalingα-stable random fields introduced in Biermé et al.

(Stoch Proc Appl 117(3):312–332, 2007) and fractional Brownian sheets introduced in Kamont (Probab Math Stat 16:85–98, 1996). We establish a general uniform modulus of continuity of such fields in terms of the polar coordinates introduced in Biermé et al. (2007). Based on this, we determine the box-counting dimension and the Hausdorff dimension of the graph of a trajectory over a non-degenerate cubeI ⊂R^d.

Keywords Fractional random fields·Stable random sheets·Operator scaling· Selfsimilarity·Box-counting dimension·Hausdorff dimension

Mathematics Subject Classification (2020) Primary 60G60; Secondary 28A78 · 28A80·60G17·60G52

1 Introduction

In this paper, we consider a harmonizable operator scalingα-stable random sheet as introduced in [11]. The main idea is to combine the properties of operator scalingα- stable random fields and fractional Brownian sheets in order to obtain a more general class of random fields. Let us recall that a scalar valued random field{X(x):x∈R^d} is said to beoperator scalingfor some matrixE∈R^d×dand someH >0 if

{X(c^Ex):x∈R^d}^f= {c^{.d. H}X(x):x∈R^d} for allc>0, (1.1)

B

1 Department of Statistics, University of Klagenfurt, Universitätsstrasse 65-67, 9020 Klagenfurt, Austria

where^f=^.d.means equality of all finite-dimensional marginal distributions and, as usual, c^E = _∞

k=0(logc)^k

k! E^k is the matrix exponential. These fields can be regarded as an anisotropic generalization of self-similar random fields (see, e.g., [8]), whereas the fractional Brownian sheet {BH1,...,Hd(x) : x ∈ R^d}with Hurst indices 0 <

H1, . . . ,Hd < 1 can be seen as an anisotropic generalization of the well-known fractional Brownian field (see, e.g., [13]) and satisfies the scaling property

{BH₁,...,H_d(c1x1, . . . ,cdxd):x =(x1, . . . ,xd)∈R^d}

f= {.d. c₁^H1. . .c_d^HdBH₁,...,H_d(x):x∈R^d}

for all constantsc1, . . . ,cd >0. See [3,10,27] and the references therein for more information on the fractional Brownian sheet.

Throughout this paper, let d = m

j=1dj for somem ∈ Nand E˜j ∈ R^dj×dj, j = 1, . . . ,m be matrices with positive real parts of their eigenvalues. We define matricesE1, . . . ,Em ∈R^d×das

Ej =

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

0 0

...

0 E˜j

0 ...

0 0

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠ .

Further, we define the block diagonal matrixE ∈R^d×das

E =

m

j=1

Ej =

⎛

⎜⎝

E˜1 0 ...

0 E˜m

⎞

⎟⎠.

In analogy to the terminology in [11, Definition 1.1.1], a random field{X(x):x∈R^d} is calledoperator scaling stable random sheetif for someH1, . . . ,Hm >0 we have

{X(c^Ejx):x ∈R^d}^f= {c^{.d. Hj}X(x):x∈R^d} (1.2) for allc>0 and j =1, . . . ,m. Note that, by applying (1.2) iteratively, any operator scaling stable random sheet is also operator scaling for the matrixEand the exponent H = m

j=1Hj in the sense of (1.1). Further, note that this definition is indeed a generalization of operator scaling random fields, since form =1,d =d1andE = E1 = ˜E1 (1.2) coincides with the definition introduced in [4]. Another example of a random field satisfying (1.2) is given by the fractional Brownian sheet, where Ej =dj =1 for j =1, . . . ,min this case. Operator scaling stable random sheets have been proven to be quite flexible in modeling physical phenomena and can be

applied in order to extend the well-known Cahn–Hilliard phase-field model. See [1]

and the references therein for more information.

Random fields satisfying a scaling property such as (1.1) or (1.2) are very popular in modeling, see [14,22] and the references in [5] for some applications. Most of these fields are Gaussian. However, Gaussian fields are not always flexible for example in modeling heavy tail phenomena. For this purpose,α-stable random fields have been introduced. See [17] for a good introduction toα-stable random fields.

Using a moving average and a harmonizable representation, the authors in [4]

defined and analyzed two different classes of symmetricα-stable random fields sat- isfying (1.1). Following the outline in [4,5], these two classes were generalized to random fields satisfying (1.2) in [11]. The fields constructed in [4] have stationary increments, i.e., they satisfy

{X(x+h)−X(h):x∈R^d}^f= {^.d. X(x):x∈R^d} for allh∈R^d.

This property has been proven to be quite useful in studying the sample path properties.

However, the property of stationary increments is no more true for the fields constructed in [11]. The absence of this property is one of the challenging difficulties we face in determining results about their sample paths.

Another main tool in studying sample paths of operator scaling stable random sheets are polar coordinates with respect to the matrices Ej,j =1, . . . ,m, introduced in [16] and used in [4,5]. If{X(x):x ∈R^d}is an operator scaling symmetricα-stable random sheet withα=2, using (1.2), one can write the variance ofX(x),x∈R^d, as

E[X²(x)] =τE(x)^2HE[X² lE(x)

], whereH =_m

j=1HjandτE(x)is the radial part ofxwith respect toEandlE(x)is its polar part. Therefore, if the random field has stationary increments in the Gaussian case information about the behavior of the polar coordinates

τ_E˜j(x),l_E˜

j(x)

contains information about the sample path regularity. This property also holds in the stable caseα ∈ (0,2). Moreover, this also remains to be true for operator scaling random sheets which do not have stationary increments but satisfy a slightly weaker property, see Corollary3.3below.

This paper is organized as follows. In Sect.2, we introduce the main tools we need for the study in this paper. Section2.1is devoted to a spectral decomposition result from [16]. Section2.2is about the change to polar coordinates with respect to scaling matrices and we establish a relation between the radiusτE(x)and the radiiτ_E˜j(x), 1 ≤ j ≤ m, in Lemma2.2below. In Sect.3, we present the results in [11] about the existence of harmonizable and moving average representations of operator scaling α-stable random sheets. Here, we will only focus on a harmonizable representation.

Moreover, we prove that these random sheets fulfill a generalized type of modulus of continuity, which is deduced by showing the applicability of results in [5,6]. Based on this and generalizing a combination of methods used in [2,4,5,24], in Sect.4we present our results on the Hausdorff dimension and box-counting dimension of the graph of harmonizable operator scaling stable random sheets.

2 Preliminaries

2.1 Spectral Decomposition

Let A ∈ R^d×d be a matrix withp distinct positive real parts of its eigenvalues 0<

a1<· · ·<apfor somep≤d. Factor the minimal polynomial ofAinto f1, . . . , fp, where all roots of fi have real part equal toai, and defineVi =Ker

fi(A) . Then, by [16, Theorem 2.1.14],

R^d=V1⊕ · · · ⊕Vp

is a direct sum decomposition, i.e. we can write anyx∈R^duniquely as x=x1+ · · · +xp

forxi ∈ Vi, 1≤i ≤ p. Further, we can choose an inner product onR^dsuch that the subspacesV1, . . . ,Vpare mutually orthogonal. Throughout this paper, for anyx∈R^d we will choosex = x,x ^1/2as the corresponding Euclidean norm. In view of our methods this will entail no loss of generality, since all norms are equivalent.

2.2 Polar Coordinates

We now recall the results about the change to polar coordinates used in [4,5]. As before, let A ∈ R^d×d be a matrix with distinct positive real parts of its eigenvalues 0 <a1 <· · · < ap for some p ≤ d. According to [4, Sect. 2] there exists a norm · AonR^dsuch that for the unit sphere SA = {x ∈R^d : xA =1}the mapping A:(0,∞)×SA→R^d\ {0}defined byA(r, θ)=r^Aθis a homeomorphism. To be more precise, the norm · Ais defined by

xA= 1

0

t^Axdt

t , x∈R^d. (2.1)

Thus, we can write anyx∈R^d\ {0}uniquely as

x=τA(x)^AlA(x), (2.2)

whereτA(x) > 0 is called the radial part ofx with respect to AandlA(x) ∈ {x ∈ R^d:τA(x)=1}is called the direction. It is clear thatτA(x)→ ∞asx → ∞and τA(x)→0 asx →0. Further, one can extendτA(·)continuously toR^d by setting τA(0)=0. Note that, by (2.2), it is straightforward to see thatτA(·)satisfies

τA(c^Ax)=c·τA(x) for all c>0. Such functions are calledA-homogeneous.

Let us recall a result about bounds on the growth rate ofτA(·)in terms ofa1, . . . ,ap

established in [4, Lemma 2.1].

Lemma 2.1 Letε >0be small enough. Then, there exist constants K1, . . . ,K4>0 such that

K1x^a11+ε≤τA(x)≤K2x^{a p1−ε} for all x withτA(x)≤1, and

K3x^{a p1−ε}≤τA(x)≤K4x^a11+ε for all x withτA(x)≥1.

We remark that the bounds on the growth rate ofτA(·)have been improved in [5, Proposition 3.3], but the bounds given in Lemma2.1suffice for our purposes.

The following Lemma will be needed in the next section in order to give an upper bound on the modulus of continuity.

Lemma 2.2 Let E,E˜1, . . . ,E˜mbe as above. Then, there exists a constant C ≥1such that

C⁻¹

m

j=1

τ_E˜j(xj)≤τE(x)≤C

m

j=1

τ_E˜j(xj)

for any x=(x1, . . . ,xm)∈R^d1× · · · ×R^dm =R^d.

Proof LetR^d¯j := {0} × · · · × {0} ×R^dj × {0} × · · · × {0} ⊂R^d, 1 ≤ j ≤m, be a subspace and note that

R^d =R^d¯1⊕ · · · ⊕R^d¯m

is a direct sum decomposition with respect toE. Throughout, writex=(x1, . . . ,xm)=

¯

x1+ · · · + ¯xmwith respect to this decomposition. From [15, Lemma 2.2], we have for somec≥1

1 c

m

i=1

τE(¯xi)≤τE(x)≤c

m

i=1

τE(x¯i).

It remains to proveτE(x¯i)=τ_E˜i(xi)for 1≤i ≤m. Without loss of generality assume i =1 and for simplicity in this proof let us assume thatm=2. Thus, for any vector x∈R^dlet us writex=(x1,x2)∈R^d1×R^d2.Note that by definition

(x₁,0)=τE(x₁,0)^El_E(x₁,0)=

τE(x₁,0)^E˜1l_E(x₁,0)1, τE(x₁,0)^E˜2l_E(x₁,0)2

=

τE(x1,0)^E˜1lE(x1,0)1,0 ,

where we used the notationlE(x)=

lE(x)1,lE(x)2

∈R^d1 ×R^d2.But on the other hand one can write

x1=τ_E˜1(x1)^E˜1l_E˜

1(x1) yielding that

τ_E˜1(x1)^E˜1l_E˜

1(x1)=τE(x1,0)^E˜1lE(x1,0)1. Further noting that

lE(x1,0)=

lE(x1,0)1,lE(x1,0)2

=

lE(x1,0)1,0

and taking into account the definition of the norm · _E˜1 given in (2.1) we obtain l_E˜1(x1)_E˜1 =1=

lE(x1,0)1,0 E=

1 0

t^E

lE(x1,0)1,0 dt t

t

= ₁

0

t^E˜1lE(x1,0)1dt t

t = lE(x1,0)1_E˜1

Thus, by the uniqueness of the representation we have τ_E˜1(x1) = τE(x1,0) and l_E˜

1(x1)=lE(x1,0)1as desired. This concludes the proof.

Corollary 2.3 Let E,E˜1, . . . ,E˜m be as above. Then, there exists a constant C ≥ 1 such that

C⁻¹

m

j=1

τ_E˜j(xj)^H ≤τE(x)^H

for any H >0and x =(x1, . . . ,xm)∈R^d1× · · · ×R^dm =R^d.

3 Harmonizable Operator Scaling Random Sheets

We consider harmonizable operator scaling stable random sheets defined in [11] and present some related results established in [11]. Most of these will also follow from the results derived in [4,5]. Throughout this paper, for j =1, . . . ,massume that the real parts of the eigenvalues ofE˜jare given by 0<a₁^j <· · ·<ap^j_j for somepj ≤dj. Let qj =trace(E˜j). Suppose thatψj :R^dj → [0,∞)are continuousE˜^T_j-homogeneous functions, which means according to [4, Definition 2.6] that

ψj(c^E˜Tjx)=cψj(x) for allc>0.

Moreover, we assume thatψj(x)=0 forx =0. See [4,5] for various examples of such functions.

Let 0 < α ≤ 2 andW_α(dξ)be a complex isotropic symmetricα-stable random measure onR^dwith Lebesgue control measure (see [17, Chaper 6.3]).

Theorem 3.1 For any vector x∈R^dlet x =(x1, . . . ,xm)∈R^d1 × · · · ×R^dm =R^d. The random field

X_α(x)=Re

R^d

m j=1

(e^ixj,ξj −1)ψj(ξj)^{−Hj−q jα} W_α(dξ), x∈R^d (3.1)

exists and is stochastically continuous if and only if Hj ∈(0,a₁^j)for all j =1, . . . ,m.

Proof This result has been proven in detail in [11], but it also follows as an easy consequence of [4, Theorem 4.1]. By the definition of stable integrals (see [17]), X_α(x)exists if and only if

α(x)=

R^d

m j=1

|e^ixj,ξj −1|^αψj(ξj)^−αHj−qjdξ <∞,

but this is equivalent to

αj(x)=

R^{d j} |e^ixj,ξj −1|^αψj(ξj)^−αHj−qjdξj <∞,

for all j = 1, . . . ,m. Since, in [4, Theorem 4.1], it is shown that _α^j(x)is finite if and only if Hj ∈ (0,a₁^j)the statement follows, see [11] for details. The stochastic continuity can be deduced similarly as a consequence of [4, Theorem 4.1].

Note that from (3.1) it follows thatX_α(x)=0 for allx =(x1, . . . ,xm)∈R^d1×

· · · ×R^dm =R^dsuch thatxj =0 for at least one j ∈ {1, . . . ,m}.

The following result has been established in [11, Corollary 4.2.1]. The proof is carried out as the proof of [4, Corollary 4.2 (a)] via characteristic functions of stable integrals and by noting thatc^Ejx=(x1, . . . ,xj−1,c^E˜jxj,xj+1, . . . ,xm)for allc>0 andx=(x1, . . . ,xm)∈R^d1× · · · ×R^dm =R^d.

Corollary 3.2 Under the conditions of Theorem3.1, the random field{X_α(x):x∈R^d} is operator scaling in the sense of(1.2), that is, for any c>0

{X(c^Ejx):x ∈R^d}^f= {^.d. c^HjX(x):x∈R^d}. (3.2) As we shall see below, fractional Brownian sheets fall into the class of random fields given by (3.1). It is known that a fractional Brownian sheet does not have stationary increments. Thus, in general, a random field given by (3.1) does not possess stationary increments. But it satisfies a slightly weaker property, as the following statement shows.

Corollary 3.3 Under the conditions of Theorem3.1, for any h∈R^dj, j =1, . . . ,m {Xα(x1, . . . ,x_j−1,xj+h,x_j+1, . . . ,xm)−X_α(x1, . . . ,x_j−1,h,x_j+1, . . . ,xm):x∈R^d}

f.d.= {X_α(x):x∈R^d},

where we used the notation x=(x1, . . . ,xm)∈R^d1× · · · ×R^dm =R^d.

Proof This result has been established in [11, Corollary 4.2.2] and is proven similarly

to [4, Corollary 4.2 (b)].

As an easy consequence of the results in this paper, we will derive global Hölder critical exponents of the random fields defined in (3.1). Following [7, Definition 5], β ∈(0,1)is said to be the Hölder critical exponent of the random field{X(x):x ∈ R^d}, if there exists a modificationX^∗of X such that for anys ∈ (0, β)the sample paths ofX^∗satisfy almost surely a uniform Hölder condition of orderson any compact setI ⊂R^d, i.e., there exists a positive and finite random variableZ such that almost surely

|X^∗(x)−X^∗(y)| ≤Zx−y^s for allx,y∈I, (3.3) whereas, for anys∈(β,1), (3.3) almost surely fails.

Let us now state our main result of this section. Note that under the assumption Hj <a¹_j and up to considering matricesE¯j = ^E_H^j_j instead ofEj in (1.2), 1≤ j ≤m, and with the observation that

c₁^jτ_E˜j(xj)^Hj ≤τE j˜ H j

(xj)≤c₂^jτ_E˜j(xj)^Hj, ∀xj ∈R^dj

for some positive and finite constantsc₁^j,c₂^j as noted in [6, Remark 5.1], without loss of generality we will assume Hj =1<a¹_j in the proof of the following statement.

We will make this assumption for notational convenience.

Proposition 3.4 Under the above assumptions and the assumption that Hj = 1 or, equivalently a₁^j >1for j =1, . . . ,m there exists a modification X_α^∗of the random field in(3.1)such that for anyε >0and any non-empty compact set Gd ⊂R^d

sup

x,y∈Gd

x=y

|X^∗_α(x)−X^∗_α(y)|

_m

j=1τ_E˜j(xj −yj)^Hj log

1+_n

j=1τ_E˜j(xj−yj)⁻¹¹

2

<∞ a.s.

ifα=2and sup

x,y∈G_d x=y

|X^∗_α(x)−X^∗_α(y)|

_m

j=1τ_E˜j(xj −yj)^Hj log

1+_n

j=1τ_E˜j(xj−yj)⁻¹_ε+¹

2+_α¹ <∞ a.s.

ifα∈(0,2), where we used the notation x =(x1, . . . ,xm)∈R^d1× · · · ×R^dm =R^d. In particular, for any0 < γ < Hj and x =(x1, . . . ,xm),y =(y1, . . . ,ym)∈ Gd

one can find a positive and finite constant C such that

|X^∗_α(x)−X_α^∗(y)| ≤C

m

j=1

τ_E˜j(xj−yj)^γ (3.4)

holds almost surely.

Proof Let us first assume thatα=2. In the following let · pdenote the p-norm for p≥1,can unspecified positive constant,Gd ⊂R^d an arbitrary compact set,r >0 andBE(r)= {x∈R^d:τE(x)≤r}. Moreover, by

dX(x,y)=E[|X2(x)−X2(y)|²]¹², x,y∈R^d,

we denote the canonical metric associated toX2. We first show forx,y∈Gdthat

dX(x,y)≤cτE(x−y). (3.5)

By the equivalence of norms one can find a constantcsuch that 1

c

m

i=1

|ui|²≤ ^m

i=1

|ui|2

≤c

m

i=1

|ui|²

for anyu ∈R^m. Further let us remark that by definition the variance of the centered Gaussian random variableX2(x)in (3.1) is given by

2(x)=E[X2(x)²] =c

R^d

m j=1

|e^ixj,ξj −1|²ψj(ξj)^−2−qjdξ.

Note that for all 1 ≤ j ≤ m andx = (x1, . . . ,xm) ∈ Gd one can find a constant 0<M(x) <∞such that

2(x1, . . . ,xj−1, θ,xj+1, . . . ,xn)≤M(x)≤ max

x∈Gd

M(x)=:M∈(0,∞),

whereθ ∈R^dj withτ_E˜j(θ)=1. Using all this and the elementary inequality

|X2(x)−X2(y)|

≤

m

i=1

|X2(x1, . . . ,xi−1,xi,yi+1, . . . ,ym)−X2(x1, . . . ,xi−1,yi,yi+1, . . . ,ym)|

with the convention that

X2(x1, . . . ,xi−1,yi,yi+1, . . . ,ym)=X2(y) fori =1 and

X2(x1, . . . ,xi−1,xi,yi+1, . . . ,ym)=X2(x) fori =mwe get for allx=(x1, . . . ,xm)andy=(y1, . . . ,ym)∈Gd

E[|X2(x)−X2(y)|²]

≤E ^m

i=1

|X2(x1, . . . ,xi−1,xi,yi+1, . . . ,ym)−X2(x1, . . . ,xi−1,yi,yi+1, . . . ,ym)|2

≤cE ^m

i=1

|X₂(x₁, . . . ,x_i−1,x_i,y_i+1, . . . ,y_m)−X₂(x₁, . . . ,x_i−1,y_i,y_i+1, . . . ,y_m)|²

=cE ^m

i=1

|X2(x1, . . . ,xi−1,xi−yi,yi+1, . . . ,ym)|2 ,

where we used Corollary3.3in the equality and the equivalence of norms in the last inequality. Using the operator scaling property and the generalized polar coordinates forxi−yi we can further get an upper estimate of the last expression by

c

m

i=1

τ_E˜i(xi−yi)²E

|X2(x1, . . . ,xi−1,l_E˜

i(xi −yi),yi+1, . . . ,ym)|²

≤cM

m

i=1

τ_E˜i(xi−yi)²≤cMτE(x−y)²,

where we used Corollary2.3with H =2 in the last inequality, which proves (3.5).

Now define an auxiliary Gaussian random fieldY = {Y(t,s):t ∈ Gd,s ∈ BE(r)}

by

Y(t,s)=X2(t+s)−X2(t), t ∈Gd,s∈ BE(r),

wherer >0 is such that BE(r)⊂Gd. Denote byDthe diameter ofGd×BE(r)in the metricdY associated withY. Then, using (3.5) it is easy to see that D ≤cr for some positive constantc. Using the latter inequality, by the arguments made in the proof of [15, Theorem 4.2] ifN(ε)denotes the smallest number of opendY-balls of radiusε >0 needed to coverGd×BE(r)we obtain that

_D

0

logN(ε)dε≤cr

log(1+r⁻¹).

Then, it follows from [21, Lemma 2.1] that for allu ≥2cr

log(1+r⁻¹)

P

sup

(t,s)∈G_d×B_E(r)|X2(t+s)−X2(t)| ≥u ≤exp

− u² 4D²

.

Therefore, by a standard Borel-Cantelli argument we conclude sup

x,y∈G_d x=y

|X^∗₂(x)−X₂^∗(y)|

τE(x−y) log

1+τE(x−y)⁻¹ <∞ a.s. (3.6) for a continuous modificationX^∗₂of X2, which by Lemma2.2is equivalent to

sup

x,y∈G_d x=y

|X_α^∗(x)−X_α^∗(y)|

m

j=1τ_E˜j(xj−yj)^Hj log

1+n

j=1τ_E˜j(xj−yj)⁻¹¹₂ <∞ a.s.

Let us now assume thatα∈(0,2). In this case, the proof is a slight modification and extension of the proof of [6, Proposition 5.1] and the idea is to check the assumptions (i), (ii) and (iii) of Proposition 4.3 of the latter reference. Throughout this proof, we letcbe a universal unspecified positive and finite constant and in the following let

f_α(u, ξ)= m j=1

(e^iuj,ξj −1)ψ_α(ξ)

with

ψ_α(ξ)= m j=1

ψj(ξj)^{−1−q jα} .

As in [6, Example 5.1] one checks that for allξ ∈R^d,ξj =0,

ψ_α(ξ)≤c m j=1

τ_E˜^T

j(ξj)^{−1−q jα} (3.7) and, in particular there exist constantsAj >0, 1≤ j ≤m, such that (3.7) holds for allξj>Aj. Forζ >0 chosen arbitrarily small we consider the functionμ˜ onR^d given byμ(ξ)˜ =m

j=1μ˜j(ξj)with

˜

μj(ξj)=

ξj +1 _α

1

ξj≤A_j+τ_E˜^T

j(ξj)^−qj|logτ_E˜^T

j(ξj)|^−1−ζ1

ξj>A_j.

We observe thatμ˜ is positive onR^d\ {0}and, similarly to the calculations made in the proof of [6, Proposition 5.1], we obtain that

R^{d j} μ˜j(ξj)dξj =c∈(0,∞), ∀1≤ j ≤m. Defineμj = ^μ˜_c^j. Moreover, note that

R^dμ(ξ)dξ˜ =c∈(0,∞).

Hence,μ= ^μ_c^˜ is well defined and now, as in the proof of [6, Proposition 5.1], we are going to check the assumptions (i), (ii) and (iii) of [6, Proposition 4.3] for

V1(u)= f_α(u, )μ()^−α1,

whereu ∈R^dandis assumed to be a random vector onR^dwith densityμ.

We choose a constantc∈(0,∞)such that

m j=1

(e^ixj,ξj −1)− m j=1

(e^iyj,ξj −1)≤ m j=1

cτE(x−y)^E˜Tjξj_E˜^T

j.

Note that this is possible. Then, it follows

|V1(x)−V1(y)| ≤ |ψα()|

m j=1

min

cρ(x,y)^E˜Tj j_E˜^T

j,1

for the quasi-metricρonR^ddefined by

ρ(x,y)=τE(x−y), ∀x,y∈R^d. Hence, we have

|V1(x)−V1(y)| ≤g

ρ(x,y),

withgdefined by

g(h, ξ)= |ψ_α(ξ)|

m j=1

min

ch^E˜Tjξj_E˜^T

j ,1 ,

so that we precisely recover assumption (i) in [6, Proposition 4.3] for the random field G=

g(h, ξ)

h∈[0,∞). Moreover, assumption (ii) immediately follows as in the proof of [6, Proposition 5.1] from the definition of the norms · _E˜^T

j and by noting that the

product of monotonic functions again is monotonic. It remains to prove assumption (iii) in [6, Proposition 4.3]. To this end we write

I(h):=E[G²(h)] =

R^d g(h, ξ)²μ(ξ)^1−2αdξ

= m j=1

R^{d j} min

ch^E˜Tjξj_E˜^T

j ,1 2

ψj(ξj)^{−2−2q jα} μj(ξj)^1−α2dξj.

Using equality (3.7) similarly as shown in the calculations made in the proof of [6, Proposition 5.1] we obtain that

R^{d j} min

ch^E˜Tjξj_E˜^T

j,1 2

ψj(ξj)^{−2−2q jα}μj(ξj)^1−α2dξj ≤ch²|log(h)|^{2(1+ζ)(α1−12)} which yields that

I(h)= m j=1

R^{d j} min

ch^E˜Tjξj_E˜^T

j,1 2

ψj(ξj)^{−2−2q jα} μj(ξj)^1−α2dξj

≤ch^2m|log(h)|^{2m(1+ζ)(α1−12)},

so that assumption (iii) in [6, Proposition 4.3] with pminstead of pis fulfilled. Fol- lowing the lines of the proof of [6, Proposition 5.1], we obtain that there exists a modificationX^∗_αof X_αsuch that

sup

x,y∈Gd

x=y

|X_α^∗(x)−X_α^∗(y)|

τE(x−y) log

1+τE(x−y)⁻¹_ε+¹

2+¹_α <∞ a.s.

for anyε > 0 and any non-empty compact setGd ⊂R^d, which by Lemma 2.2is equivalent to

sup

x,y∈G_d x=y

|X_α^∗(x)−X_α^∗(y)|

_m

j=1τ_E˜j(xj −yj) log

1+_m

j=1τ_E˜j(xj−yj)⁻¹_ε+¹

2+_α¹ <∞ a.s.

This completes the proof.

Corollary 3.5 Under the assumptions of Theorem 3.1, there exist a positive and finite random variable Z and a continuous modification of X_α such that for any s∈(0,min1≤j≤m

H_j

a_{p j}^j )the uniform Hölder condition(3.3)holds almost surely.

We remark that Corollary3.5is not a statement about critical Hölder exponents.

However, as a consequence of Theorem4.1below, we will see that any continuous version ofX_α admits min1≤j≤m

H_j

a_{p j}^j as the critical exponent.

4 Hausdorff Dimension

We now state our main result on the Hausdorff and box-counting dimension of the graph of X_α defined in (3.1). In the following, for a set B ⊂ R^d we denote by dim_BB, dim_BBand dim_HBits lower, upper box-counting and Hausdorff dimension, respectively. We refer to [9] for a definition of these objects.

Theorem 4.1 Suppose that the conditions of Theorem3.1hold. Then, for any contin- uous version of X_α, almost surely

dim_HGX_α

[0,1]^d

=dim_BGX_α

[0,1]^d

=d+1− min

1≤j≤m

Hj

ap^j_j

, (4.1)

where

GX_α

[0,1]^d

=

x,X_α(x)

:x∈ [0,1]^d is the graph of X_αover[0,1]^d.

Proof Let us choose a continuous version of X_α by Corollary3.5. From Corollary 3.5, for any 0<s <min1≤j≤m

H_j

a_{p j}^j , the sample paths ofX_α satisfy almost surely a uniform Hölder condition of orderson[0,1]^d. Thus, by ad-dimensional version of [9, Corollary 11.2] we have

dim_HGX_α

[0,1]^d

≤dim_BGX_α

[0,1]^d

≤d+1−s, a.s.

Lettings↑min1≤j≤m Hj

a_{p j}^j along rational numbers yields the upper bound in (4.1).

It remains to prove the lower bound in (4.1). Since the inequality dim_BB ≥dim_HB

holds for everyB⊂R^d(see [9, Chapter 3.1]), it suffices to show dim_HGX_α

[0,1]^d

≥d+1− min

1≤j≤m

Hj

ap^j_j

, a.s.

Further, note that, sinceQ= [¹₂,1]^d⊂ [0,1]^d, we have dim_HGX_α

[0,1]^d

≥dim_HGX_α(Q)

by monotonicity of the Hausdorff dimension. Thus, it is even enough to show that dim_HGX_α(Q)≥d+1− min

1≤j≤m

Hj

ap^jj

, a.s. (4.2)