Abstract
We consider operator scaling \(\alpha \)-stable random sheets, which were introduced in Hoffmann (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 scaling \(\alpha \)-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 cube \(I \subset {\mathbb {R}}^d\).
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1 Introduction
In this paper, we consider a harmonizable operator scaling \(\alpha \)-stable random sheet as introduced in [11]. The main idea is to combine the properties of operator scaling \(\alpha \)-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 \(\{ X(x) : x \in {\mathbb {R}}^d \}\) is said to be operator scaling for some matrix \(E \in {\mathbb {R}}^{d \times d}\) and some \(H>0\) if
where \({\mathop {=}\limits ^\mathrm{f.d.}}\) means equality of all finite-dimensional marginal distributions and, as usual, \(c^E = \sum _{k=0}^{\infty } \frac{(\log c)^k}{k!} E^k\) 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 \(\{ B_{H_1, \ldots , H_d} (x) : x \in {\mathbb {R}}^d \}\) with Hurst indices \(0<H_1, \ldots , H_d <1\) can be seen as an anisotropic generalization of the well-known fractional Brownian field (see, e.g., [13]) and satisfies the scaling property
for all constants \(c_1, \ldots , c_d >0\). See [3, 10, 27] and the references therein for more information on the fractional Brownian sheet.
Throughout this paper, let \(d= \sum _{j=1}^m d_j\) for some \(m \in {\mathbb {N}}\) and \({\tilde{E}}_j \in {\mathbb {R}}^{d_j \times d_j}\), \(j=1, \ldots , m\) be matrices with positive real parts of their eigenvalues. We define matrices \(E_1, \ldots , E_m \in {\mathbb {R}}^{d \times d}\) as
Further, we define the block diagonal matrix \(E \in {\mathbb {R}}^{d \times d}\) as
In analogy to the terminology in [11, Definition 1.1.1], a random field \(\{ X(x) : x \in {\mathbb {R}}^d \}\) is called operator scaling stable random sheet if for some \(H_1, \ldots , H_m >0\) we have
for all \(c>0\) and \(j =1, \ldots , m\). Note that, by applying (1.2) iteratively, any operator scaling stable random sheet is also operator scaling for the matrix E and the exponent \(H= \sum _{j=1}^m H_j\) in the sense of (1.1). Further, note that this definition is indeed a generalization of operator scaling random fields, since for \(m=1, d =d_1\) and \(E=E_1 = {\tilde{E}}_1\) (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 \(E_j=d_j=1\) for \(j =1, \ldots , m\) in 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, \(\alpha \)-stable random fields have been introduced. See [17] for a good introduction to \(\alpha \)-stable random fields.
Using a moving average and a harmonizable representation, the authors in [4] defined and analyzed two different classes of symmetric \(\alpha \)-stable random fields satisfying (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
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 \(E_j, j=1, \ldots , m\), introduced in [16] and used in [4, 5]. If \(\{X( x) : x \in {\mathbb {R}}^d \}\) is an operator scaling symmetric \(\alpha \)-stable random sheet with \(\alpha = 2\), using (1.2), one can write the variance of \(X( x), x \in {\mathbb {R}}^d\), as
where \(H= \sum _{j=1}^m H_j\) and \(\tau _{E} (x)\) is the radial part of x with respect to E and \(l_{E} (x)\) is its polar part. Therefore, if the random field has stationary increments in the Gaussian case information about the behavior of the polar coordinates \(\big ( \tau _{{\tilde{E}}_j} (x), l_{{\tilde{E}}_j} (x) \big )\) contains information about the sample path regularity. This property also holds in the stable case \(\alpha \in (0,2)\). 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 Corollary 3.3 below.
This paper is organized as follows. In Sect. 2, we introduce the main tools we need for the study in this paper. Section 2.1 is devoted to a spectral decomposition result from [16]. Section 2.2 is about the change to polar coordinates with respect to scaling matrices and we establish a relation between the radius \(\tau _{E} (x)\) and the radii \(\tau _{{\tilde{E}}_j} (x)\), \(1 \le j \le m\), in Lemma 2.2 below. In Sect. 3, we present the results in [11] about the existence of harmonizable and moving average representations of operator scaling \(\alpha \)-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. 4 we 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 \(A \in {\mathbb {R}}^{d \times d}\) be a matrix with p distinct positive real parts of its eigenvalues \(0< a_1< \cdots < a_p\) for some \(p \le d\). Factor the minimal polynomial of A into \(f_1, \ldots , f_p\), where all roots of \(f_i\) have real part equal to \(a_i\), and define \(V_{i}={\text {Ker}}\big ( f_{i}(A) \big )\). Then, by [16, Theorem 2.1.14],
is a direct sum decomposition, i.e. we can write any \(x \in {\mathbb {R}}^d\) uniquely as
for \(x_i \in V_i\), \(1 \le i \le p\). Further, we can choose an inner product on \({{\mathbb {R}}^d}\) such that the subspaces \(V_1, \ldots , V_p\) are mutually orthogonal. Throughout this paper, for any \(x \in {{\mathbb {R}}^d}\) we will choose \( \Vert x \Vert = \langle x,x\rangle ^{1/2}\) as 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 \(A \in {\mathbb {R}}^{d \times d}\) be a matrix with distinct positive real parts of its eigenvalues \(0< a_1< \cdots < a_p\) for some \(p \le d\). According to [4, Sect. 2] there exists a norm \(\Vert \cdot \Vert _A\) on \({{\mathbb {R}}^d}\) such that for the unit sphere \(S_A = \{ x \in {{\mathbb {R}}^d}: \Vert x \Vert _A = 1 \}\) the mapping \(\Psi _A : (0, \infty ) \times S_A \rightarrow {{\mathbb {R}}^d}\setminus \{0 \}\) defined by \(\Psi _A (r, \theta ) = r^A \theta \) is a homeomorphism. To be more precise, the norm \(\Vert \cdot \Vert _A\) is defined by
Thus, we can write any \(x \in {{\mathbb {R}}^d}\setminus \{0 \}\) uniquely as
where \(\tau _A (x) >0\) is called the radial part of x with respect to A and \(l_A(x) \in \{ x \in {{\mathbb {R}}^d}: \tau _A (x) = 1 \}\) is called the direction. It is clear that \(\tau _A (x) \rightarrow \infty \) as \(\Vert x\Vert \rightarrow \infty \) and \(\tau _A (x) \rightarrow 0\) as \(\Vert x\Vert \rightarrow 0\). Further, one can extend \(\tau _A( \cdot )\) continuously to \({{\mathbb {R}}^d}\) by setting \(\tau _A(0) = 0\). Note that, by (2.2), it is straightforward to see that \(\tau _A( \cdot )\) satisfies
Such functions are called A-homogeneous.
Let us recall a result about bounds on the growth rate of \(\tau _A( \cdot )\) in terms of \(a_1, \ldots , a_p\) established in [4, Lemma 2.1].
Lemma 2.1
Let \(\varepsilon >0\) be small enough. Then, there exist constants \(K_1, \ldots , K_4 >0\) such that
for all x with \(\tau _A(x) \le 1\), and
for all x with \(\tau _A(x) \ge 1.\)
We remark that the bounds on the growth rate of \(\tau _A (\cdot )\) have been improved in [5, Proposition 3.3], but the bounds given in Lemma 2.1 suffice 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 \(E, {\tilde{E}}_1, \ldots , {\tilde{E}}_m\) be as above. Then, there exists a constant \(C \ge 1\) such that
for any \(x = (x_1, \ldots , x_m ) \in {\mathbb {R}}^{d_1} \times \cdots \times {\mathbb {R}}^{d_m} = {{\mathbb {R}}^d}\).
Proof
Let \({\mathbb {R}}^{{\bar{d}}_j} := \{ 0\} \times \cdots \times \{0\} \times {\mathbb {R}}^{d_j} \times \{0\} \times \cdots \times \{0\} \subset {{\mathbb {R}}^d}\), \(1 \le j \le m\), be a subspace and note that
is a direct sum decomposition with respect to E. Throughout, write \(x = (x_1, \ldots , x_m ) = {\bar{x}}_1 + \cdots + {\bar{x}}_m\) with respect to this decomposition. From [15, Lemma 2.2], we have for some \(c \ge 1\)
It remains to prove \(\tau _E ({\bar{x}}_i) = \tau _{{\tilde{E}}_i} (x_i)\) for \(1 \le i \le m\). Without loss of generality assume \(i=1\) and for simplicity in this proof let us assume that \(m=2\). Thus, for any vector \(x\in {{\mathbb {R}}^d}\) let us write \(x = (x_1, x_2) \in {\mathbb {R}}^{d_1} \times {\mathbb {R}}^{d_2}.\) Note that by definition
where we used the notation \(l_E(x) = \big ( l_E(x)_1, l_E(x)_2 \big ) \in {\mathbb {R}}^{d_1} \times {\mathbb {R}}^{d_2}.\) But on the other hand one can write
yielding that
Further noting that
and taking into account the definition of the norm \(\Vert \cdot \Vert _{{\tilde{E}}_1}\) given in (2.1) we obtain
Thus, by the uniqueness of the representation we have \(\tau _{{\tilde{E}}_1} (x_1) = \tau _{E} (x_1,0)\) and \(l_{{\tilde{E}}_1} (x_1) = l_{E} (x_1,0)_1\) as desired. This concludes the proof. \(\square \)
Corollary 2.3
Let \(E, {\tilde{E}}_1, \ldots , {\tilde{E}}_m\) be as above. Then, there exists a constant \(C \ge 1\) such that
for any \(H>0\) and \(x = (x_1, \ldots , x_m ) \in {\mathbb {R}}^{d_1} \times \cdots \times {\mathbb {R}}^{d_m} = {{\mathbb {R}}^d}\).
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 \(j =1, \ldots , m\) assume that the real parts of the eigenvalues of \({\tilde{E}}_j\) are given by \(0< a_1^j< \cdots < a_{p_j}^j\) for some \(p_j \le d_j\). Let \(q_j = {{\,\mathrm{trace}\,}}({\tilde{E}}_j ) \). Suppose that \(\psi _j : {\mathbb {R}}^{d_j} \rightarrow [0, \infty ) \) are continuous \({\tilde{E}}_j^T\)-homogeneous functions, which means according to [4, Definition 2.6] that
Moreover, we assume that \(\psi _j(x) \ne 0\) for \(x \ne 0\). See [4, 5] for various examples of such functions.
Let \(0 < \alpha \le 2\) and \(W_{\alpha } (d \xi )\) be a complex isotropic symmetric \(\alpha \)-stable random measure on \({{\mathbb {R}}^d}\) with Lebesgue control measure (see [17, Chaper 6.3]).
Theorem 3.1
For any vector \(x \in {{\mathbb {R}}^d}\) let \(x= (x_1, \ldots , x_m) \in {\mathbb {R}}^{d_1} \times \cdots \times {\mathbb {R}}^{d_m} = {{\mathbb {R}}^d}\). The random field
exists and is stochastically continuous if and only if \(H_j \in (0, a_1^j)\) for all \(j=1, \ldots , m\).
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]), \(X_\alpha (x)\) exists if and only if
but this is equivalent to
for all \(j=1, \ldots , m\). Since, in [4, Theorem 4.1], it is shown that \(\Gamma _\alpha ^j (x)\) is finite if and only if \(H_j \in (0, a_1^j)\) the statement follows, see [11] for details. The stochastic continuity can be deduced similarly as a consequence of [4, Theorem 4.1]. \(\square \)
Note that from (3.1) it follows that \(X_\alpha (x) = 0\) for all \(x= (x_1, \ldots , x_m) \in {\mathbb {R}}^{d_1} \times \cdots \times {\mathbb {R}}^{d_m} = {{\mathbb {R}}^d}\) such that \(x_j = 0\) for at least one \(j \in \{ 1, \ldots , m \}\).
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 that \(c^{E_j} x = (x_1, \ldots , x_{j-1}, c^{{\tilde{E}}_j} x_j, x_{j+1}, \ldots , x_m )\) for all \(c>0\) and \(x = (x_1, \ldots , x_m ) \in {\mathbb {R}}^{d_1} \times \cdots \times {\mathbb {R}}^{d_m} = {{\mathbb {R}}^d}\).
Corollary 3.2
Under the conditions of Theorem 3.1, the random field \(\{ X_\alpha (x) : x \in {{\mathbb {R}}^d}\}\) is operator scaling in the sense of (1.2), that is, for any \(c>0\)
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 Theorem 3.1, for any \(h \in {\mathbb {R}}^{d_j}\), \(j =1, \ldots , m\)
where we used the notation \(x = (x_1, \ldots , x_m ) \in {\mathbb {R}}^{d_1} \times \cdots \times {\mathbb {R}}^{d_m} = {{\mathbb {R}}^d}\).
Proof
This result has been established in [11, Corollary 4.2.2] and is proven similarly to [4, Corollary 4.2 (b)]. \(\square \)
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], \(\beta \in (0,1)\) is said to be the Hölder critical exponent of the random field \(\{ X(x) : x \in {{\mathbb {R}}^d}\}\), if there exists a modification \(X^*\) of X such that for any \(s \in (0, \beta )\) the sample paths of \(X^*\) satisfy almost surely a uniform Hölder condition of order s on any compact set \(I \subset {{\mathbb {R}}^d}\), i.e., there exists a positive and finite random variable Z such that almost surely
whereas, for any \(s \in ( \beta ,1)\), (3.3) almost surely fails.
Let us now state our main result of this section. Note that under the assumption \(H_j < a^1_j\) and up to considering matrices \({\bar{E}}_j = \frac{E_j}{H_j}\) instead of \(E_j\) in (1.2), \(1 \le j \le m\), and with the observation that
for some positive and finite constants \(c_1^j, c_2^j\) as noted in [6, Remark 5.1], without loss of generality we will assume \(H_j=1<a^1_j\) 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 \(H_j=1\) or, equivalently \(a_1^j>1\) for \(j=1, \ldots , m\) there exists a modification \(X_\alpha ^*\) of the random field in (3.1) such that for any \(\varepsilon >0\) and any non-empty compact set \(G_d \subset {{\mathbb {R}}^d}\)
if \(\alpha =2\) and
if \(\alpha \in (0,2)\), where we used the notation \(x=(x_1, \ldots , x_m) \in {\mathbb {R}}^{d_1} \times \cdots \times {\mathbb {R}}^{d_m} = {{\mathbb {R}}^d}\). In particular, for any \(0<\gamma <H_j\) and \(x=(x_1, \ldots , x_m), y=(y_1, \ldots , y_m) \in G_d\) one can find a positive and finite constant C such that
holds almost surely.
Proof
Let us first assume that \(\alpha =2\). In the following let \(\Vert \cdot \Vert _p\) denote the p-norm for \(p \ge 1\), c an unspecified positive constant, \(G_d \subset {{\mathbb {R}}^d}\) an arbitrary compact set, \(r>0\) and \(B_E (r) = \{ x \in {{\mathbb {R}}^d}: \tau _E (x) \le r\}\). Moreover, by
we denote the canonical metric associated to \(X_2\). We first show for \(x,y \in G_d\) that
By the equivalence of norms one can find a constant c such that
for any \(u \in {\mathbb {R}}^m\). Further let us remark that by definition the variance of the centered Gaussian random variable \(X_2(x)\) in (3.1) is given by
Note that for all \(1\le j \le m\) and \(x=(x_1, \ldots , x_m) \in G_d\) one can find a constant \(0<M (x)< \infty \) such that
where \(\theta \in {{\mathbb {R}}}^{d_j}\) with \(\tau _{{\tilde{E}}_j} (\theta ) = 1\). Using all this and the elementary inequality
with the convention that
for \(i=1\) and
for \(i=m\) we get for all \(x= (x_1, \ldots , x_m)\) and \(y= (y_1, \ldots , y_m) \in G_d\)
where we used Corollary 3.3 in the equality and the equivalence of norms in the last inequality. Using the operator scaling property and the generalized polar coordinates for \(x_i-y_i\) we can further get an upper estimate of the last expression by
where we used Corollary 2.3 with \(H=2\) in the last inequality, which proves (3.5). Now define an auxiliary Gaussian random field \( Y = \{ Y(t,s) : t \in G_d , s \in B_E(r)\}\) by
where \(r>0\) is such that \( B_E(r) \subset G_d\). Denote by D the diameter of \(G_d \times B_E(r)\) in the metric \(d_Y\) associated with Y. Then, using (3.5) it is easy to see that \(D \le c r \) for some positive constant c. Using the latter inequality, by the arguments made in the proof of [15, Theorem 4.2] if \(N(\varepsilon )\) denotes the smallest number of open \(d_Y\)-balls of radius \(\varepsilon >0\) needed to cover \(G_d \times B_E(r)\) we obtain that
Then, it follows from [21, Lemma 2.1] that for all \(u \ge 2cr \sqrt{ \log (1+r^{-1}) }\)
Therefore, by a standard Borel-Cantelli argument we conclude
for a continuous modification \(X_2^*\) of \(X_2\), which by Lemma 2.2 is equivalent to
Let us now assume that \(\alpha \in (0,2)\). 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 let c be a universal unspecified positive and finite constant and in the following let
with
As in [6, Example 5.1] one checks that for all \(\xi \in {{\mathbb {R}}^d}\), \(\xi _j \ne 0\),
and, in particular there exist constants \(A_j >0\), \(1\le j \le m\), such that (3.7) holds for all \(\Vert \xi _j\Vert > A_j\). For \(\zeta >0\) chosen arbitrarily small we consider the function \({\tilde{\mu }}\) on \({{\mathbb {R}}^d}\) given by \({\tilde{\mu }} (\xi ) = \prod _{j=1}^m {\tilde{\mu }}_j (\xi _j)\) with
We observe that \({\tilde{\mu }}\) is positive on \({{\mathbb {R}}^d}\setminus \{ 0\}\) and, similarly to the calculations made in the proof of [6, Proposition 5.1], we obtain that
Define \(\mu _j = \frac{\tilde{\mu _j}}{c}\). Moreover, note that
Hence, \(\mu = \frac{{\tilde{\mu }}}{c}\) 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
where \(u \in {{\mathbb {R}}^d}\) and \(\Xi \) is assumed to be a random vector on \({{\mathbb {R}}^d}\) with density \(\mu \).
We choose a constant \(c \in (0, \infty )\) such that
Note that this is possible. Then, it follows
for the quasi-metric \(\rho \) on \({\mathbb {R}}^{d}\) defined by
Hence, we have
with g defined by
so that we precisely recover assumption (i) in [6, Proposition 4.3] for the random field \({\mathcal {G}} = \big ( g(h, \xi ) \big ) _{h \in [0, \infty )}\). Moreover, assumption (ii) immediately follows as in the proof of [6, Proposition 5.1] from the definition of the norms \(\Vert \cdot \Vert _{{\tilde{E}}^T_j}\) 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
Using equality (3.7) similarly as shown in the calculations made in the proof of [6, Proposition 5.1] we obtain that
which yields that
so that assumption (iii) in [6, Proposition 4.3] with pm instead of p is fulfilled. Following the lines of the proof of [6, Proposition 5.1], we obtain that there exists a modification \(X_\alpha ^*\) of \(X_\alpha \) such that
for any \(\varepsilon >0\) and any non-empty compact set \(G_d \subset {{\mathbb {R}}^d}\), which by Lemma 2.2 is equivalent to
This completes the proof. \(\square \)
Corollary 3.5
Under the assumptions of Theorem 3.1, there exist a positive and finite random variable Z and a continuous modification of \(X_\alpha \) such that for any \(s \in (0, \min _{1 \le j \le m} \frac{H_j}{a_{p_j}^j} )\) the uniform Hölder condition (3.3) holds almost surely.
We remark that Corollary 3.5 is not a statement about critical Hölder exponents. However, as a consequence of Theorem 4.1 below, we will see that any continuous version of \(X_\alpha \) admits \(\min _{1 \le j \le m} \frac{H_j}{a_{p_j}^j}\) as the critical exponent.
4 Hausdorff Dimension
We now state our main result on the Hausdorff and box-counting dimension of the graph of \(X_\alpha \) defined in (3.1). In the following, for a set \(B \subset {{\mathbb {R}}^d}\) we denote by \(\underline{\dim _{{\mathcal {B}}}} B\), \(\overline{\dim _{{\mathcal {B}}}} B\) and \(\dim _{{\mathcal {H}}} B\) its 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 Theorem 3.1 hold. Then, for any continuous version of \(X_\alpha \), almost surely
where
is the graph of \(X_\alpha \) over \([0,1]^d\).
Proof
Let us choose a continuous version of \(X_\alpha \) by Corollary 3.5. From Corollary 3.5, for any \(0< s < \min _{1 \le j \le m} \frac{H_j}{a_{p_j}^j}\), the sample paths of \(X_\alpha \) satisfy almost surely a uniform Hölder condition of order s on \([0,1]^d\). Thus, by a d-dimensional version of [9, Corollary 11.2] we have
Letting \(s \uparrow \min _{1 \le j \le m} \frac{H_j}{a_{p_j}^j}\) along rational numbers yields the upper bound in (4.1).
It remains to prove the lower bound in (4.1). Since the inequality
holds for every \(B \subset {{\mathbb {R}}^d}\) (see [9, Chapter 3.1]), it suffices to show
Further, note that, since \(Q = [ \frac{1}{2} , 1]^d \subset [0,1]^d\), we have
by monotonicity of the Hausdorff dimension. Thus, it is even enough to show that
We will show this by combining the methods used in [2, 4, 5]. From now on, without loss of generality, we will assume that
Let \(\gamma > 1\). Following the argument in [4, Theorem 5.6], in view of the Frostman criterion [9, Theorem 4.13 (a)], it suffices to show that
in order to obtain \(\dim _{{\mathcal {H}}} G_{X_\alpha } (Q) \ge \gamma \) almost surely.
Using the characteristic function of the symmetric \(\alpha \)-stable random field \(X_\alpha \), as in the proof of [5, Proposition 5.7], it can be shown that there is a constant \(C_1 >0\) such that
where
Combining this with Theorem 4.2 below we get
for some \({\tilde{C}}_1>0\) and \(Q_1 = [ \frac{1}{2} , 1]^{d_1}\). With this inequality the assertion readily follows from the proof of [4, Theorem 5.6]. \(\square \)
The following Theorem is crucial for proving Theorem 4.1 and its proof is based on [23, Theorem 1]. See also [24,25,26]. Let us remark that a similar method of the following proof has been applied in [24, Theorem 3.4] for certain \(\alpha \)-stable random fields if \(1\le \alpha \le 2\). In the following, we are able to extend this method for \(0<\alpha <1\) and, in particular this shows that the statement of [24, Theorem 3.5] can be formulated for \(0<\alpha <1\) as well.
Theorem 4.2
There exists a constant \(C_4 >0\), depending on \(H_1, \ldots , H_m, q_1, \ldots , q_m\) and d only, such that for all \(x = (x_1, \ldots x_m), y =(y_1, \ldots y_m) \in [\frac{1}{2} , 1)^{d_1} \times \cdots \times [\frac{1}{2} , 1)^{d_m}\) we have
where \(\tau _{{\tilde{E}}_1} ( \cdot )\) is the radial part with respect to \({\tilde{E}}_1\).
Proof
Throughout this proof, we fix \(x = (x_1, \ldots , x_m)\), \(y =(y_1, \ldots y_m) \in [\frac{1}{2} , 1)^{d_1} \times \cdots \times [\frac{1}{2} , 1)^{d_m}\). We will show that
for some constant \(C>0\) independent of x and y and \(r = \tau _{{\tilde{E}}_1} (x_1 - y_1)\). Without loss of generality we will assume that \(r>0\), since for \(r=0\) (4.3) always holds. By definition, we have
Now, for every \(j=1, \ldots , m\) we consider a so-called bump function \(\delta _j \in C^{\infty } ({\mathbb {R}}^{d_j} )\) with values in [0, 1] such that \(\delta _j (0) = 1\) and \(\delta _j\) vanishes outside the open ball
for
where \(\varepsilon >0\) is some (sufficiently) small number and \(K_1^j , \ldots , K_4^j\) are the suitable constants derived from Lemma 2.1 corresponding to the matrix \({\tilde{E}}_j\). The choice of the constant \(K_j >0\) will be clear later in this proof. Let \({\hat{\delta }}_j\) be the Fourier transform of \(\delta _j\). It can be verified that \({\hat{\delta }}_j \in C^\infty ( {\mathbb {R}}^{d_j} )\) as well and that \({\hat{\delta }}_j (\lambda _j)\) decays rapidly as \(\Vert \lambda _j \Vert \rightarrow \infty \). By the Fourier inversion formula, we have
for all \(s_j \in {\mathbb {R}}^{d_j}\). Let \(\delta _1^r (s_1) = \frac{1}{r^{q_1}} \delta _1 \big ( (\frac{1}{r} )^{{\tilde{E}}_1} s_1 \big ) .\) Then, by a change of variables in (4.5), for all \(s_1 \in {\mathbb {R}}^{d_1}\) we obtain
Using Lemma 2.1 and the fact that \(\tau _{{\tilde{E}}_1} ( \cdot )\) is \({\tilde{E}}_1\)-homogeneous, it is straightforward to see that \(\tau _{{\tilde{E}}_j} (x_j) \ge K_j\), \(\tau _{{\tilde{E}}_1} \big ( (\frac{1}{r})^{{\tilde{E}}_1}(x_1 - y_1) \big ) \ge K_1\) and \(\tau _{{\tilde{E}}_1} \big ( (\frac{1}{r})^{{\tilde{E}}_1}x_1 \big ) \ge K_1\) for \(r=\tau _{{\tilde{E}}_1} (x_1-y_1)>0\). Therefore, we have \(\delta _1^r (x_1) = 0\), \(\delta _1^r (x_1 - y_1) = 0\) and \(\delta _j (x_j) = 0\) for all \(j =2, \ldots , m\). Hence, combining this with (4.5) and (4.6) it follows that
For every \(\alpha \in (0,2)\) we can choose \(k \in {\mathbb {N}}\) such that \(k \alpha \ge 1\) and let \(\beta ' >1\) be the constant such that \(\frac{1}{k\alpha } + \frac{1}{\beta '} = 1\). We first show that
For \(\lambda = (\lambda _1, \ldots , \lambda _m) \in {\mathbb {R}}^{d_1} \times \cdots \times {\mathbb {R}}^{d_m}\), let
and note that, since \(| e^{it} -1 |^2 = 2 - 2 \cos t \le 4\) for all \(t \in {\mathbb {R}}\), it follows that
From this, we obtain
Now, using (4.8), by Hölder’s inequality and (4.4) we have
where \(\tilde{{\tilde{C}}} >0\) is a constant, which only depends on \(H_1, \ldots , H_m, q_1, \ldots , q_m, k, \alpha , d\) and \(\delta \). It is clear that (4.3) follows from (4.7) and (4.9). This finishes the proof of the Theorem. \(\square \)
As an immediate consequence of Theorem 4.1, we obtain the following.
Corollary 4.3
Let the assumptions of Theorem 3.1 hold. Then any continuous version of \(X_\alpha \) admits \(\min _{1 \le j \le m} \frac{H_j}{a_{p_j}^j}\) as the Hölder critical exponent.
Remark 4.4
Let \(\alpha = 2, d_j = {\tilde{E}}_j = 1\) for all \(j=1, \ldots , m\) and consider the function \(\psi (\xi _j) = | \xi _j |\) for all \(\xi _j \in {\mathbb {R}}\). Clearly, \(\psi _j\) is a homogeneous function and satisfies \(\psi _j (\xi _j ) \ne 0\) for all \(\xi _j \ne 0\). Thus, by Theorem 3.1, we can define
for all \(0<H_j<1, j=1, \ldots , d\) and the statement of Theorem 4.1 becomes
Further, up to a multiplicative constant, the random field \(X_2\) is a fractional Brownian sheet with Hurst indices \(H_1, \ldots , H_d\) (see [10]). Thus, Theorem 4.1 can be seen as a generalization of [2, Theorem 1.3]. Further, as noted above, for \(m=1, d=d_1\) and \(E=E_1 = {\tilde{E}}_1\) the random field \(X_\alpha \) given by (3.1) coincides with the random field in [4, Theorem 4.1] and the statement of Theorem 4.1 becomes
which is the statement of [4, Theorem 5.6] for \(\alpha =2\) and [5, Proposition 5.7] for \(\alpha \in (0,2)\). We finally remark that Theorem 4.1 can be proven similarly, if we replace \([0,1]^d\) in (4.1) by any other compact cube of \({{\mathbb {R}}^d}\).
We finally remark that the Hausdorff dimension only depends on solely one index \(\min _{1 \le j \le m} \frac{H_j}{a_{p_j}^j}\), whereas in higher space dimensions all indices are relevant and the corresponding results in more general dimensions can be found in [18]. See also [19, 20] for related results.
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Acknowledgements
The author would like to thank an anonymous referee and associate editor for some comments on earlier versions of the manuscript. A large part of this work has been written during the author’s employment at Heinrich-Heine-Universität Düsseldorf. This work has been supported by Deutsche Forschungsgemeinschaft (DFG) under Grant KE1741/ 6-1.
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Sönmez, E. Sample Path Properties of Generalized Random Sheets with Operator Scaling. J Theor Probab 34, 1279–1298 (2021). https://doi.org/10.1007/s10959-020-01045-6
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DOI: https://doi.org/10.1007/s10959-020-01045-6
Keywords
- Fractional random fields
- Stable random sheets
- Operator scaling
- Selfsimilarity
- Box-counting dimension
- Hausdorff dimension