Home Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
Article Open Access

Small values and functional laws of the iterated logarithm for operator fractional Brownian motion

  • Wensheng Wang EMAIL logo and Jingshuang Dong
Published/Copyright: August 21, 2024
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

The multivariate Gaussian random fields with matrix-based scaling laws are widely used for inference in statistics and many applied areas. In such contexts, interests are often Hölder regularities of spatial surfaces in any given direction. This article analyzes the almost sure sample function behavior for operator fractional Brownian motion, including multivariate fractional Brownian motion. We obtain the estimations of small ball probability and the strongly locally nondeterministic for operator fractional Brownian motion in any given direction. By applying these estimates, we obtain Chung type laws of the iterated logarithm for operator fractional Brownian motion. Our results show that the precise Hölder regularities of these spatial surfaces are completely determined by the real parts of the eigenvalues of self-similarity exponent and the covariance matrix at time point 1.

Keywords: operator self-similarity; operator fractional Brownian motion; small ball probability; Chung type laws of the iterated logarithm; strong local nondeterminism
MSC 2010: 60F15; 60G17; 60G22; 60G60

1 Introduction

Let X = { X ( t ) = ( X 1 ( t ) , … , X p ( t ) ) , t ∈ R } be an operator fractional Brownian motion with exponent D , that is, X is a mean zero Gaussian process in R p , has stationary increments and is operator self-similar with exponent D , X ( 0 ) = 0 a.s. We will use the following definition for operator self-similarity, which corresponds to that of operator self-similar random fields of Sato [1]. An R p -valued random field X = { X ( t ) , t ∈ R } is said to be operator self-similar if there exists an D ∈ L ( R p ) , where L ( R p ) is the set of linear operators on R p , such that for all c > 0 ,

(1) X ( c ⋅ ) = d c D X ( ⋅ ) ,

where X = d Y means the processes X and Y have the same finite dimensional distributions and c D = ∑ k = 0 ∞ 1 k ! ( ln c ) k D k .

An operator fractional Brownian motion has been introduced in the seminal papers of Laha and Rohatgi [2], Hudson and Mason [3], Maejima and Mason [4], and Didier and Pipiras [5] as extensions to the class of fractional Brownian motion. If p = 1 , X is the standard fractional Brownian motion. If the operator self-similar exponent D = diag ( H 1 , … , H p ) is a diagonal operator, where H i ∈ ( 0 , 1 ) for all 1 ≤ i ≤ p , then X is referred to multivariate fractional Brownian motion. See Lavancier et al. [6,7], Coeurjolly et al. [8], and Li et al. [9] for more information about multivariate fractional Brownian motion.

By using the operator self-similarity and the stationarity of the increments, Lavancier et al. [6] first studied the cross-covariance structure of multivariate fractional Brownian motion. In fact, their results do not rely on the Gaussian hypothesis. Amblard et al. [10] further studied the cross-covariance and parameterized this covariance structure in a simpler way.

Firstly, the i th component of the multivariate fractional Brownian motion is a fractional Brownian motion with exponent H i ∈ ( 0 , 1 ) , 1 ≤ i ≤ p . The cross covariances are given in the following proposition.

Proposition 1.1

[6] The cross covariances of the multivariate fractional Brownian motion satisfy the following representation, for all ( i , j ) ∈ { 1 , … , p } 2 , i ≠ j ,

  1. If H i + H j ≠ 1 , there exist σ i > 0 , σ j > 0 , ( ρ i , j , η i , j ) ∈ [ − 1 , 1 ] × R with ρ i , j = ρ j , i = corr ( X i ( 1 ) , X j ( 1 ) ) and η i , j = − η j , i such that

    (2) E [ X i ( s ) X j ( t ) ] = σ i σ j 2 { ( ρ i , j + η i , j sign ( s ) ) ∣ s ∣ H i + H j + ( ρ i , j − η i , j sign ( t ) ) ∣ t ∣ H i + H j − ( ρ i , j − η i , j sign ( t − s ) ) ∣ t − s ∣ H i + H j } .

  2. If H i + H j = 1 , there exist σ i > 0 , σ j > 0 , ( ρ ˜ i , j , η ˜ i , j ) ∈ [ − 1 , 1 ] × R with ρ ˜ i , j = ρ ˜ j , i = corr ( X i ( 1 ) , X j ( 1 ) ) and η ˜ i , j = − η ˜ j , i such that

    (3) E [ X i ( s ) X j ( t ) ] = σ i σ j 2 { ρ ˜ i , j ( ∣ s ∣ + ∣ t ∣ − ∣ s − t ∣ ) + η ˜ i , j ( t ln ∣ t ∣ − s ln ∣ s ∣ − ( t − s ) ln ∣ t − s ∣ ) } .

Remark 1.1

As mentioned in [6], coefficients ρ i , j , ρ ˜ i , j , η i , j , η ˜ i , j in (2) and (3) depend on ( H i , H j ) . Note that ( ( s + 1 ) H − s H − 1 ) ⁄ ( 1 − H ) → s ln ∣ s ∣ − ( s + 1 ) ln ∣ s + 1 ∣ as H → 1 . If the cross-covariance functions are continuous with respect to ( H i , H j ) , then (3) can be deduced from (2) by letting H i + H j → 1 . We have: as H i + H j → 1

ρ i , j ∼ ρ ˜ i , j and ( 1 − H i − H j ) η i , j ∼ η ˜ i , j .

This convergence result can suggest a reparameterization of coefficients η i , j in ( 1 − H i − H j ) η i , j .

The multivariate models evoke several applications where matrix-based scaling laws are expected to appear, such as in long range dependent time series [11,12] and queueing systems [13,14]. Like fractional Brownian motion in the univariate setting, operator fractional Brownian motion is a natural selection for constructing estimators for operator self-similarity process, because it is Gaussian and closely connected with stationary fractional process (see [2,9] for more information about operator self-similar processes). The fractal nature for operator fractional Brownian motion such as the Hausdorff dimension of the image and graph, and spatial surface properties such as hitting probabilities, transience, and the characterization of polar sets were studied by Mason and Xiao [15]. The moduli of continuity for operator fractional Brownian motion were investigated by Wang [16].

The purpose of this article is to investigate Hölder regularities of spatial surfaces for operator fractional Brownian motion in any given direction. We obtain the estimations of small ball probability and the prediction error for operator fractional Brownian motion in any given direction. By applying these estimates, we investigate its small values and prove its Chung type laws of the iterated logarithm. A Chung type law of the iterated logarithm for multivariate fractional Brownian motion is derived from it as a consequence. Our results show that Hölder regularities for operator fractional Brownian motion in any given direction are completely determined by the self-similarity exponent and the covariance matrix at time point 1. Our results extend the related results for fractional Brownian motion obtained by Monrad and Rootzén [17].

Let U = ( u i j ) be a real invertible p × p matrix U such that Cov ( X ( 1 ) ) = U U * . In fact, it is symmetric and holds whenever Cov ( X ( 1 ) ) is invertible. For any vector θ ∈ Γ ≔ R p \ { 0 } , define φ : ( 0 , ∞ ) → ( 0 , ∞ ) by φ ( x ) = φ ( x , θ ) = ‖ ( x D U ) * θ ‖ , where A * denotes the transpose of the matrix or vector A , and ‖ ⋅ ‖ is the Euclidean norm on R p . For t 0 ∈ R , h ∈ R + , θ ∈ Γ and compact rectangle T ⊂ R , we denote by M ( t 0 , h ) the local modulus of continuity of X ( t ) on t 0 in direction θ ,

(4) M ( t 0 , h ) = M ( X ; t 0 , h , θ ) = sup s ∈ T , ∣ s ∣ ≤ h ∣ ⟨ X ( t 0 + s ) − X ( t 0 ) , θ ⟩ ∣ ,

write L h = ln ( h ∨ e ) and L 2 h = ln ln ( h ∨ e 2 ) . This note is devoted to establishing the following:

Theorem 1.1

Let X = { X ( t ) , t ∈ R } be an operator fractional Brownian motion in R p with exponent D, and let a 1 and a d be the minimal and maximal real parts of eigenvalues of D, respectively. If 0 < a 1 , a d < 1 , and detCov ( X ( 1 ) ) > 0 , then for any vector θ ∈ Γ , there exists a positive and finite constant K 1 , 1 = K 1 , 1 ( θ ) such that for any compact set T ⊂ R and any t 0 ∈ T ,

(5) liminf h → 0 + f h M ( t 0 , h ) = K 1 , 1 a.s. ,

where

(6) f h = 1 φ ( h ⁄ L 2 h ) .

Remark 1.2

  • Equation (5) implies that for any fixed t 0 ∈ R and θ ∈ Γ , there exists an event Ω 0 = Ω 0 ( t 0 , θ ) ⊂ Ω of probability zero with the following two properties:

    1. for any ω ∉ Ω 0 and any sequence h 1 > h 2 > … , there exist a subsequence h k j = h k j ( t 0 , θ , ω ) such that

      f h k j M ( t 0 , h k j , ω ) → K 1 , 1 ,

    2. for any ε > 0 and ω ∉ Ω 0 , there exists a positive constant h 0 = h 0 ( ε , t 0 , θ , ω ) such that for all 0 < h < h 0 and some x ∈ [ 0 , 1 ] ,

      f h ∣ ⟨ X ( t 0 + h x , ω ) − X ( t 0 , ω ) , θ ⟩ ∣ ≥ K 1 , 1 − ε .

  • In the case of p = 1 and D = H ∈ ( 0 , 1 ) (i.e., the case of fractional Brownian motion), equation (5) with f h = h − H ( L 2 h ) H , H ∈ ( 0 , 1 ) , is obtained by Monrad and Rootzén [17]. So Theorem 1.1 extends the related results for fractional Brownian motion.

  • It is well known that K 1 , 1 = π ⁄ 8 in (5) for Brownian motion (i.e., the case of p = 1 and D = 1 ⁄ 2 ). See, for example, [18].

  • In the case of p = 1 and D = H ∈ ( 0 , 1 ) (i.e., the case of fractional Brownian motion), the constant K 1 , 1 = κ H in (5), where κ is referred to as the small ball constant of fractional Brownian motion. In the case H = 1 ⁄ 2 , it is known that κ = π 2 ⁄ 8 . In the case H ≠ 1 ⁄ 2 , the value of κ is not known. Shao [19] proved that 0.08 ⁄ ( 2 H ) ≤ κ ≤ 10 ⁄ 2 H for 0 < H < 1 ⁄ 2 . Li and Linde [20] proved that κ can be represented analytically as follows:

    (7) κ = ( − inf ϑ > 0 ϑ 1 ⁄ H log P ( sup 0 ≤ t ≤ 1 ∣ W t H ∣ ≤ ϑ ⁄ a H ) ) H ,

    where a H is a positive constant and W H is the Riemann-Liouville fractional Brownian motion. See Wang and Xiao [21] for further information.

The rest of our article is organized as follows. In Section 2, we first define spectral index function according to Jordan decomposition’s theory and give several basic properties of the operator norm and exponential operators, then we obtain small ball probability estimation for operator fractional Brownian motion by using strong local nondeterminism. In Section 3, we first obtain a zero-one law for operator fractional Brownian motion, then we establish Chung type and functional laws of the iterated logarithm for operator fractional Brownian motion.

Throughout this article, we use the notations f t ∼ g t if lim f t ⁄ g t = 1 , f t ≍ g t if there exits a constant K > 0 such that K − 1 ≤ liminf f t ⁄ g t ≤ limsup f t ⁄ g t ≤ K . An unspecified finite and positive constant will be denoted by K , which may not be the same in each occurrence. More specific constants in Section i will be denoted by K i , 1 , K i , 2 , …

2 Preliminaries

2.1 Spectral index function and exponential operators

It follows from the Jordan decomposition’s theorem (see [22] or [23] p. 129) that every D ∈ L ( R p ) has a real canonical form; i.e., there exists a real invertible p × p matrix P such that E = P − 1 D P is composed of diagonal blocks which are either Jordan cell matrix of the form

v 0 … 0 1 v ⋱ ⋮ ⋮ ⋱ ⋱ 0 0 … 1 v

with v a real eigenvalue of D or blocks of the form

(8) Λ 0 ⋯ ⋯ 0 I 2 Λ ⋱ ⋱ ⋮ 0 ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ 0 0 ⋯ 0 I 2 Λ with Λ = a − b b a and I 2 = 1 0 0 1 ,

where the complex numbers a ± i b ( b ≠ 0 ) are complex conjugated eigenvalues of D .

Denote by v j , j = 1 , … , d , d ≤ p , the eigenvalues of D and 0 < a j = ℜ ( v j ) < 1 the real part of v j , j = 1 , … , d . Then there exist J 1 , … , J d and a real p × p invertible matrix P such that

D = P J 1 0 ⋯ 0 0 J 2 0 ⋮ ⋮ ⋱ ⋱ 0 0 ⋯ 0 J d P − 1 ,

where each J j is either a Jordan cell matrix or a block of the form (8). We can assume that each J j is associated with the eigenvalue v j of D and that

0 < a 1 < a 2 < … < a d < 1 .

If v j ∈ R , J j is a Jordan cell matrix of size l ˜ j = l j ∈ N \ { 0 } . If λ j ∈ C \ R , J j is a block of the form (8) of size l ˜ j = 2 l j ∈ 2 N \ { 0 } . Then for any t > 0 ,

t D = P t J 1 0 ⋯ 0 0 t J 2 0 ⋮ ⋮ ⋱ ⋱ 0 0 ⋯ 0 t J d P − 1 .

We denote by ( e 1 , … , e p ) the canonical basis of R p and set f j = P e j for every j = 1 , … , p . Hence, ( f 1 , … , f p ) is a basis of R p . For all j = 1 , … , d , let

V j = span f k ; ∑ i = 1 j − 1 l ˜ i + 1 ≤ k ≤ ∑ i = 1 j l ˜ i .

Then, each V j is a D -invariant set and R p = V 1 ⊕ … ⊕ V d is a direct sum decomposition of R p into D -invariant subspaces. We may write D = D 1 ⊕ … ⊕ D d , where D i : V i → V i and every eigenvalue of D i has real part equal to a i . The matrix for D in an appropriate basis is then block-diagonal with d blocks, the i th block corresponding to the matrix for D i .

Let λ i = a i − 1 so that λ 1 > … > λ d . Let λ ( θ ) : R p \ { 0 } → { λ 1 , … , λ d } be the spectral index function, that is,

(9) λ ( θ ) = λ i = 1 ⁄ a i for all θ ∈ L i \ L i − 1 ,

where L i = V 1 ⊕ … ⊕ V i and V 1 , … , V d is the spectral decomposition R p = V 1 ⊕ … ⊕ V d with respect to D . Choose an inner product ⟨ ⋅ , ⋅ ⟩ on R p such that V i ⊥ V j for i ≠ j , and let ‖ x ‖ = ⟨ x , x ⟩ be the associated Euclidean norm. The operator norm of the linear operator A on L ( R p ) is defined by

‖ A ‖ = sup { ‖ A x ‖ : ‖ x ‖ = 1 } .

We first state several useful facts about the operator norm and exponential operators whose proofs are easy (see, e.g., [22] or [24] for their proofs) and will be used to our proofs.

  1. ‖ x ‖ ⁄ ‖ A − 1 ‖ ≤ ‖ A x ‖ ≤ ‖ A ‖ ⋅ ‖ x ‖ for all A ∈ L ( R p ) and all x ∈ R p ;

  2. ‖ A B ‖ ≤ ‖ A ‖ ⋅ ‖ B ‖ for all A , B ∈ L ( R p ) ;

  3. If A ∈ L ( R p ) and s , t > 0 , then s A t A = ( s t ) A ;

  4. If A ∈ L ( R p ) and t > 0 , then t − A = ( 1 ⁄ t ) A = ( t A ) − 1 ;

  5. If A B = B A and t > 0 , then t A t B = t A + B ;

  6. If t ≥ 1 , then K 2 , 1 t a 1 − ε ≤ ‖ t D ‖ ≤ K 2 , 2 t a d + ε for any 0 < ε < a 1 ;

  7. If 0 < t < 1 , then K 2 , 3 t a d + ε ≤ ‖ t D ‖ ≤ K 2 , 4 t a 1 − ε for any 0 < ε < a 1 ;

Let U = ( u i j ) be a real invertible p × p matrix U such that Cov ( X ( 1 ) ) = U U * . In fact, it is symmetric and holds whenever Cov ( X ( 1 ) ) is invertible. For any vector θ ∈ Γ ≔ R p \ { 0 } , define φ : ( 0 , ∞ ) → ( 0 , ∞ ) by φ ( x ) = φ ( x , θ ) = ‖ ( x D U ) * θ ‖ , where A * denotes the transpose of the matrix or vector A .

Lemma 2.1

Let θ ∈ Γ . Then, for any ε > 0 , there exist positive and finite constants K 2 , i = K 2 , i ( ε , θ ) , i = 5 , … , 8 , such that for any s > 0 and h ≥ 1 ,

(10) K 2 , 5 h 1 λ ( θ ) − ε ≤ φ ( h s ) φ ( s ) ≤ K 2 , 6 h 1 λ ( θ ) + ε ,

and for any s > 0 and h < 1 ,

(11) K 2 , 7 h 1 λ ( θ ) + ε ≤ φ ( h s ) φ ( s ) ≤ K 2 , 8 h 1 λ ( θ ) − ε .

Proof

When h = 1 , it is clearly valid and does not need to be proven. The proofs of both cases h > 1 and h < 1 are similar, so we only prove the case h > 1 . For any θ ∈ Γ , there exists a unique 1 ≤ i ≤ d such that θ ∈ L i \ L i − 1 , where L i = V 1 ⊕ … ⊕ V i and V 1 , … , V d is the spectral decomposition R p = V 1 ⊕ … ⊕ V d with respect to D . Moreover, for any θ ∈ L i \ L i − 1 , there exist θ j ∈ V j , θ j ≠ 0 , 1 ≤ j ≤ i , such that θ = θ 1 + … + θ i and ( t D U ) * θ = ( t D 1 U 1 ) * θ 1 + … + ( t D i U i ) * θ i , where D 1 , … , D d is the spectral decomposition of D and U 1 , … , U d is the spectral decomposition of U . Then, for any h ≥ 1 ,

φ 2 ( h , θ ) φ 2 ( h , θ i ) = ‖ ( h D U ) * θ ‖ 2 ‖ ( h D i U i ) * θ i ‖ 2 = 1 + ∑ j = 1 i − 1 ‖ ( h D j U j ) * θ j ‖ 2 ‖ ( h D i U i ) * θ i ‖ 2 .

Noting that every eigenvalue of D j has real part equal to a j , by Facts (i), (ii), and (vi), we have that for any ε > 0 and 1 ≤ j ≤ i ,

(12) ‖ ( h D j U j ) * θ j ‖ ≤ ‖ ( h D j ) * θ j ‖ ⋅ ‖ U j ‖ ≤ K 2 , 9 ‖ h D j ‖ ≤ K 2 , 10 h a j + ε

and

(13) ‖ ( h D j U j ) * θ j ‖ ≥ ‖ ( h D j ) * θ j ‖ ⁄ ‖ U j − 1 ‖ ≥ K 2 , 11 ‖ θ j ‖ ⁄ ‖ ( 1 ⁄ h ) D j ‖ ≥ K 2 , 12 h a j − ε .

Since ε > 0 is arbitrary and a j < a i for all 1 ≤ j ≤ i − 1 , we have φ ( h , θ ) ≍ φ ( h , θ i ) . Thus, by Facts (ii) and (vi), for any ε > 0 ,

φ ( h s ) φ ( s ) ≍ φ ( h s , θ i ) φ ( s , θ i ) = ‖ ( h D i ) * ( s D i U i ) * θ i ‖ ‖ ( s D i U i ) * θ i ‖ ≤ ‖ ( h D i ) * ‖ ≤ K 2 , 13 h a i + ε .

Similarly to the aforementioned inequality, we have

φ ( s ) φ ( h s ) ≤ ‖ ( ( 1 ⁄ h ) D i ) * ‖ ≤ K 2 , 14 h − a i + ε .

The proof is completed.□

We need the following lemma, which is proved in [25]; see also [26].

Lemma 2.2

Consider a function ψ such that N d ( S , ε ) ≤ ψ ( ε ) for all ε > 0 . Assume that for some constant K > 0 and all ε > 0 , we have

ψ ( ε ) ⁄ K ≤ ψ ε 2 ≤ K ψ ( ε ) .

Then

(14) P ( sup s , t ∈ S ∣ Z ( s ) − Z ( t ) ∣ ≤ u ) ≥ exp ( − K ψ ( u ) ) .

2.2 Strong local nondeterminism

Now we start to construct a moving average representation of operator fractional Brownian motion.

Lemma 2.3

Let D ∈ L ( R p ) be a linear operator with 0 < a 1 , a d < 1 . For t ∈ R , define

(15) X ( t ) = ∫ − ∞ 0 ( t − x ) D − 1 2 I − ( − x ) D − 1 2 I B ( d x ) + ∫ 0 t ( t − x ) D − 1 2 I B ( d x ) ,

where I ∈ L ( R p ) is the identity operator and { B ( s ) , − ∞ < s < ∞ } is p-dimensional standard Brownian motion and i.i.d. components. Then the random field X = { X ( t ) , t ∈ R } is an operator fractional Brownian motion with exponent D. Furthermore, X is isotropic in the sense that for every t ∈ R ,

(16) X ( t ) = d ∣ t ∣ D X ( 1 ) ,

and X has a version with continuous sample paths almost surely.

Proof

The proof is similar to that for the stochastic integral representation of operator fractional Brownian motion given in Theorem 3.1 in [15], we omit the details. The proof is completed.□

The following result establishes the strongly locally nondeterministic for operator fractional Brownian motion in any given direction θ ∈ Γ .

Lemma 2.4

Let X = { X ( t ) , t ∈ R } be an operator fractional Brownian motion in R p with exponent D. If 0 < a 1 , a d < 1 , and detCov ( X ( 1 ) ) > 0 , then for any vector θ ∈ Γ , there exists a positive and finite constant K 2 , 15 = K 2 , 15 ( θ ) such that all 0 < h < h 0 and all 0 < t < h 0 − h with some h 0 > 0 ,

(17) Var ( ⟨ X ( t + h ) , θ ⟩ ∣ ⟨ X ( s ) , θ ⟩ : 0 ≤ s ≤ t ) ≥ K 2 , 15 φ 2 ( h ) .

Proof

From the representation (15), it easily follows that if { X ( t ) } is an operator fractional Brownian motion with exponent D , then

Var ( ⟨ X ( t + h ) , θ ⟩ ∣ ⟨ X ( s ) , θ ⟩ : 0 ≤ s ≤ t ) ≥ Var ∫ t t + h ⟨ ( t + h − x ) D − 1 2 I B ( d x ) , θ ⟩

(18) = Var ∫ t t + h ⟨ ( t + h − x ) D * − 1 2 I θ , B ( d x ) ⟩ = ∫ t t + h ( t + h − x ) D * − 1 2 I θ 2 d x = ∫ 0 h ( h − x ) D * − 1 2 I θ 2 d x .

It follows from Facts (v), (vi), and (vii) that φ ( h ) ≤ ‖ U ‖ ‖ ( h D ) * θ ‖ and ‖ ( h − x ) D * − 1 2 I θ ‖ ≥ ‖ ( h − x ) D * θ ‖ ⁄ ‖ ( h − x ) 1 2 I ‖ . Thus, by Lemma 2.3,

(19) ∫ 0 h φ − 2 ( h ) ( h − x ) D * − 1 2 I θ 2 d x ≥ ∫ 0 h ‖ ( h − x ) D * θ ‖ 2 ‖ U ‖ 2 ( h − x ) 1 2 I 2 ‖ ( h D ) * θ ‖ 2 d x ≥ ∫ 0 h ( 1 − x ⁄ h ) 2 λ ( θ ) + 2 ε ‖ U ‖ 2 ( h − x ) d x = ∫ 0 h ( h − x ) 2 λ ( θ ) − 1 + 2 ε ‖ U ‖ 2 h 2 λ ( θ ) + 2 ε d x ≥ K 2 , 16 .

Combining (18) and (19), we obtain (17). The proof is completed.□

2.3 Small ball probability

We establish the following estimation of small ball probability of spatial surfaces for operator fractional Brownian motion in any given direction θ ∈ Γ .

Proposition 2.1

Let X = { X ( t ) , t ∈ R } be an operator fractional Brownian motion in R p with exponent D. If 0 < a 1 , a d < 1 , and detCov ( X ( 1 ) ) > 0 , then for any θ ∈ Γ , there exit positive and finite constants K 2 , 17 = K 2 , 17 ( θ ) and K 2 , 18 = K 2 , 18 ( θ ) such that for any compact set T ⊂ R , t 0 ∈ T , h > 0 , and x ∈ ( 0 , 1 ) ,

(20) exp − K 2 , 17 h Ψ ( x 2 ) ≤ P ( M ( t 0 , h ) ≤ x ) ≤ exp − K 2 , 18 h Ψ ( x 2 ) ,

where M ( t 0 , h ) = M ( t 0 , h , θ ) = sup s ∈ T , ∣ s ∣ ≤ h ∣ ⟨ X ( t 0 + s ) − X ( t 0 ) , θ ⟩ ∣ denotes the local modulus of continuity of X ( t ) on t 0 in direction θ , Ψ ( x ) = inf { y : φ 2 ( y ) > x } is the right-continuous inverse function of φ 2 .

Proof

Since Cov ( X ( 1 ) ) is invertible, there exists a real invertible p × p matrix U such that Cov ( X ( 1 ) ) = U U * . By the operator self-similarity, for every h ∈ R \ { 0 } ,

(21) Cov ( X ( h ) ) = Cov ( ∣ h ∣ D X ( 1 ) ) = ∣ h ∣ D U ( ∣ h ∣ D U ) * .

We denote the matrix ∣ h ∣ D U by U h . Then, for h ∈ R \ { 0 } , U h − 1 X ( h ) is normal random variables in R p with mean 0 and covariance matrix I p . Thus, for all x ∈ R ,

(22) P ( φ − 1 ( h ) ⟨ X ( t + h ) − X ( t ) , θ ⟩ ≤ x ) = P ( φ − 1 ( h ) ⟨ X ( h ) , θ ⟩ ≤ x ) = P ( φ − 1 ( h ) ⟨ U h U h − 1 X ( h ) , θ ⟩ ≤ x ) = P ( φ − 1 ( h ) ⟨ U h − 1 X ( h ) , θ ¯ h ⟩ ≤ x ) = P ( ⟨ U h − 1 X ( h ) , θ ˜ h ⟩ ≤ x ) ,

where θ ¯ h = ( U h ) * θ and θ ˜ h = ‖ ( U h ) * θ ‖ − 1 ( U h ) * θ is an unit vector in R p . Noting that ⟨ U h − 1 X ( h ) , θ ˜ h ⟩ is a standard normal random variable, (22) implies that φ − 1 ( h ) ⟨ X ( t + h ) − X ( t ) , θ ⟩ is a standard normal random variable. Thus,

(23) E [ ∣ ⟨ X ( t + h ) − X ( t ) , θ ⟩ ∣ 2 ] = φ 2 ( h ) .

Equip S = [ 0 , h ] with the canonical metric

(24) d ( s , t ) = ‖ ⟨ X ( s ) − X ( t ) , θ ⟩ ‖ 2 , s , t ∈ S ,

and denote by N d ( S , ε ) the smallest number of d -balls of radius ε > 0 needed to cover S . Then it is easy to see that for all ε ∈ ( 0 , 1 ) ,

(25) N d ( S , ε ) ≤ K 2 , 19 h Ψ ( ε 2 ) .

Moreover, it follows from Lemma 2.1 that Ψ has the doubling property, i.e., K 2 , 20 Ψ ( ε ) ≤ Ψ ( ε ⁄ 2 ) ≤ K 2 , 21 Ψ ( ε ) . Hence, the lower bound in (25) follows from Lemma 2.2.

The proof of the upper bound in (20) is based on an argument in [17]. For any integer n ≥ 2 , we choose n points t n , i ∈ [ 0 , 1 ] , where t n , i = i h ⁄ n , i ∈ { 1 , … , n } . Then,

(26) P ( M ( t 0 , h ) ≤ x ) ≤ P ( max 1 ≤ i ≤ n ∣ ⟨ X ( t n , i ) , θ ⟩ ∣ ≤ x ) .

By Anderson’s inequality for Gaussian measures and Lemma 2.4, we derive the following upper bound for the conditional probabilities

(27) P ( X ( t n , i ) ≤ x ∣ X ( t n , j ) , 1 ≤ j ≤ i − 1 ) ≤ Φ K 2 , 22 x φ ( n − 1 h ) ,

where Φ ( x ) is the distribution function of a standard normal random variable. It follows from (26) and (27) that

(28) P ( M ( t 0 , h ) ≤ x ) ≤ Φ K 2 , 23 x φ ( n − 1 h ) n .

By taking n to be the smallest integer h [ Ψ ( x 2 ) ] − 1 , we obtain the upper bound in (20).□

3 Results

3.1 Zero-one laws for operator fractional Brownian motion

We establish the following zero-one laws for operator fractional Brownian motion to have Chung’s law of the iterated logarithm, which may be of independent interest.

Lemma 3.1

Let X = { X ( t ) , t ∈ R } be an operator fractional Brownian motion in R p with exponent D. If 0 < a 1 , a d < 1 , and detCov ( X ( 1 ) ) > 0 , then for any vector θ ∈ Γ , there exists a constant 0 ≤ K = K ( θ ) ≤ ∞ such that for any compact set T ⊂ R and any t 0 ∈ T ,

(29) liminf h → 0 + f h M ( t 0 , h ) = K a.s. ,

where M ( t 0 , h ) is defined in (4) and f h is defined in (6).

Proof

Let m be a scattered Gaussian random measure on R with Lebesgue measure l as its control measure; that is, { m ( A ) , A ∈ ℰ } is a centered Gaussian process on ℰ = { E ⊂ R : l ( E ) < ∞ } with covariance function

E [ m ( E ) m ( F ) ] = l ( E ∩ F ) .

Let m 1 , … , m p be p independent copies of m , and define

m ( A ) = ( m 1 ( A ) , … , m p ( A ) ) .

Then, we consider a version of operator fractional Brownian motion

(30) X ( t ) = ∫ R ( 1 − cos ( t x ) ) 1 ∣ x ∣ D + I ⁄ 2 d m ( x ) + ∫ R sin ( t x ) 1 ∣ x ∣ D + I ⁄ 2 d m ′ ( x ) ,

where m ′ is an independent copy of m . This stochastic integral representation of operator fractional Brownian motion is given in [15].

Let Ω 1 ≔ O ( 0 , 1 ) ⊂ R and for n ≥ 2 , Ω n ≔ O ( 0 , n ) \ O ( 0 , n − 1 ) ⊂ R such that Ω 1 , Ω 2 , … , are mutually disjoint, where the following notation is used: O ( x , r ) = { y ∈ R : ∣ x − y ∣ ≤ r } . For n ≥ 1 and t ∈ R , let

(31) Z n ( t ) ≔ ∫ Ω n ( 1 − cos ( t x ) ) 1 ∣ x ∣ D + I ⁄ 2 d m ( x ) + ∫ Ω n sin ( t x ) 1 ∣ x ∣ D + I ⁄ 2 d m ′ ( x ) ,

Then Z n = { Z n ( t ) , t ∈ R } , n = 1 , 2 , … , are independent Gaussian fields. By (31), we express

X ( t ) = ∑ n = 1 ∞ Z n ( t ) , t ∈ R .

Equip S = [ 0 , 1 ] with the canonical metric

d Z n ( s , t ) = d Z n ( s , t , θ ) = ‖ ⟨ Z n ( s ) − Z n ( t ) , θ ⟩ ‖ 2 , s , t ∈ T ,

and denote by N ( d Z n , S , ε ) the smallest number of d Z n -balls of radius ε > 0 needed to cover S .

It follows from (31) that

⟨ Z n ( s ) − Z n ( t ) , θ ⟩ = ∫ Ω n ( cos ( t x ) − cos ( s x ) ) 1 ∣ x ∣ D * + I ⁄ 2 θ , d m ( x ) + ∫ Ω n ( sin ( s x ) − sin ( t x ) ) 1 ∣ x ∣ D * + I ⁄ 2 θ , d m ′ ( x ) .

Thus,

(32) d Z n ( s , t ) = 2 ∫ Ω n ( 1 − cos ( ( t − s ) x ) ) 1 ∣ x ∣ D * + I ⁄ 2 θ 2 d x 1 ⁄ 2 ≤ ∣ t − s ∣ ∫ Ω n ∣ x ∣ 2 1 ∣ x ∣ D * + I ⁄ 2 θ 2 d x 1 ⁄ 2 ≕ ∣ t − s ∣ K n , s , t ∈ R .

To obtain the last inequality, in the integral, we bound 1 − cos ( t x ) by ∣ t ∣ 2 ∣ x ∣ 2 ⁄ 2 . Then, by (32) and Theorem 4.1 in [27], we have

(33) limsup h → 0 + sup t , t + s ∈ T : ∣ s ∣ ≤ h ∣ ⟨ Z n ( t + s ) − Z n ( t ) , θ ⟩ ∣ τ ( 0 , h ) ≤ K 3 , 1 a.s. ,

where τ ( 0 , h ) = ∣ h ∣ L h . Put

X M ( t ) = ∑ n = 1 M Z n ( t ) , t ∈ T .

It follows from Facts (i)–(vii) and 0 < a 1 < 1 that

h L h f h ≤ K 3 , 2 h 1 − a 1 + ε → 0

as h tends to zero. This, together with (33), yields that

lim h → 0 + sup s ∈ T : ∣ s ∣ ≤ h f h ∣ ⟨ X M ( t 0 + s ) − X M ( t 0 ) , θ ⟩ ∣ = 0 a.s.

Therefore, the random variable

liminf h → 0 + f h M ( t 0 , h )

is measurable with respect to the tail field of { Z n } n = 1 ∞ and hence is constant almost surely. This implies that (29) holds. The proof is completed.□

3.2 Chung’s laws of the iterated logarithm for spatial surfaces

In this section, we shall prove Theorem 1.1 and establish Chung’s law of the iterated logarithm for operator fractional Brownian motion. Before proving the theorem, we first obtain Chung’s laws of the iterated logarithm for multivariate fractional Brownian motion as consequences of Theorem 1.1.

Denote the standard basis of R p by ( e 1 , … , e p ) . By choosing θ = e i and using Theorem 1.1, we obtain the following result about Chung’s law of the iterated logarithm for the components X i ( i = 1 , … , p ) of X .

Corollary 3.1

Let X = { X ( t ) , t ∈ R } be an operator fractional Brownian motion in R p with exponent D. If 0 < a 1 , a d < 1 , and detCov ( X ( 1 ) ) > 0 , then for every i = 1 , … , p , there exists a positive and finite constant K 3 , 3 = K 3 , 3 ( i ) such that for any compact set T ⊂ R and any t 0 ∈ T ,

(34) liminf h → 0 + f i , h M i ( t 0 , h ) = K 3 , 3 a.s. ,

where M i ( t 0 , h ) = M i ( t 0 , h , θ ) = sup s ∈ T , ∣ s ∣ ≤ h ∣ X i ( t 0 + s ) − X i ( t 0 ) ∣ denotes the uniform modulus of continuity of the ith component of X ( t ) on t 0 , and

f i , h = 1 ‖ ( ( h ⁄ L 2 h ) D U ) * e i ‖ .

Remark 3.1

By making use of (4.6) and (4.8) in [15], (34) implies that there exists a constant p i ≥ 1 such that for any compact set T ⊂ R , t 0 ∈ T and every i = 1 , … , p ,

(35) liminf h → 0 + ( L 2 h ) a i h a i ( L h ) p i − 1 M i ( t 0 , h ) ≤ K 3 , 4 a.s. ,

where a i is defined in Section 2.

By choosing D = diag { H 1 , … , H p } in Corollary 3.1, where H i ∈ ( 0 , 1 ) for all 1 ≤ i ≤ p , as an immediate consequence of Corollary 3.1, we have the following Chung’s law of the iterated logarithm for the multivariate fractional Brownian motion, which may be of independent interest.

Proposition 3.1

Let X = { X ( t ) , t ∈ R } be a multivariate fractional Brownian motion in R p with exponent D = diag { H 1 , … , H p } , and U = ( u i j ) be a real invertible p × p matrix U such that Cov ( X ( 1 ) ) = U U * . Then, for any vector θ ∈ Γ , there exists a positive and finite constant K 3 , 5 = K 3 , 5 ( θ ) such that for any compact set T ⊂ R and any t 0 ∈ T ,

(36) liminf h → 0 + ( L 2 h ) H k h H k M ( t 0 , h ) = K 3 , 5 θ k u k 1 2 + … + u k p 2 a.s. ,

where k = argmin { H 1 , … , H p } , and for any compact set T ⊂ R , t 0 ∈ T and every i = 1 , … , p ,

(37) liminf h → 0 + ( L 2 h ) H i h H i M i ( t 0 , h ) = K 3 , 3 u i 1 2 + … + u i p 2 a.s. ,

where K 3 , 3 is given as in (34).

Remark 3.2

(36) implies that the minimum growth rate of multivariate fractional Brownian motion in any given direction θ is determined by the minimum of { H 1 , … , H p } . In addition, arg min ( H 1 , … , H p ) determines the constant on the right-hand side of (36). Although as stated in Section 1, the i th component of the multivariate fractional Brownian motion is a fractional Brownian motion with exponent H i ∈ ( 0 , 1 ) , 1 ≤ i ≤ p , (37) implies that the minimum growth rate of the i th coordinate direction of multivariate fractional Brownian motion depends on corresponding covariance matrix and hence the interrelationship between all directions.

Proof of Theorem 1.1

Throughout, it is sufficient to consider h -values which make the iterated logarithm positive and t 0 = 0 . Put M ( h ) = M ( 0 , h ) . We first show that

(38) liminf h → 0 + f h M ( h ) ≥ K 3 , 6 a.s.

Let ε > 0 and γ > 1 , and for n = 1 , 2 , … put h n = γ − n and β n = φ ( K 2 , 18 ( 1 + ε ) − 1 h n ⁄ L 2 h n ) . Then, by (20),

∑ ∞ P ( M ( h n ) ≤ β n ) ≤ K ∑ ∞ ( log γ n ) − ( 1 + ε ) < ∞ ,

where the sums are over all n large enough to make n log γ > 1 and β n < 1 . Hence, by the Borel-Cantelli lemma, M ( h n ) ≥ β n for all n greater than some n 0 = n 0 ( ω ) . Further, for n ≥ n 0 and h n ≤ h < h n − 1 ,

M ( h ) ≥ M ( h n ) ≥ β n = f h − 1 ( f h β n ) .

Hence, by Lemma 2.1, (38) holds.

Next, we prove that

(39) liminf h → 0 + f h M ( h ) ≤ K 3 , 7 a.s.

Let ε ∈ ( 0 , 1 ) and q > 1 be arbitrary real number. This time we choose

h n = e − n q , J n = k e n q , γ n = φ ( K 2 , 17 ( 1 − ε ) − 1 h n ⁄ L 2 h n ) .

Define the process Y n ( t ) ≔ Y ( t , J n − 1 , J n ) by

(40) Y ( t , J n − 1 , J n ) = ∫ ∣ x ∣ ∈ ( J n − 1 , J n ) ( 1 − cos ( t x ) ) 1 ∣ x ∣ D + I ⁄ 2 d m ( x ) + ∫ ∣ x ∣ ∈ ( J n − 1 , J n ) sin ( t x ) 1 ∣ x ∣ D + I ⁄ 2 d m ′ ( x ) ,

and denote Y ˜ n ( t ) ≔ X ( t ) − Y n ( t ) . Clearly, by (30), X ( t ) = Y n ( t ) + Y ˜ n ( t ) , and for every n = 1 , 2 , … , Y n ( ⋅ ) has stationary increments and Y n ( ⋅ ) , n = 1 , 2 , … are independent due to the virtue of independent increments of m .

For simplify notation, put M ¯ ( h ) = M ( Y ; 0 , h , θ ) and M ˜ ( h ) = M ( Y ˜ ; 0 , h , θ ) , where M is defined in Lemma 3.1. For any ε ∈ ( 0 , 1 ) , put

G n = { M ( h n ) ≤ γ n } , G ¯ n = { M ¯ ( h n ) ≤ ( 1 − ε ) γ n } and G ˜ n = { M ˜ ( h n ) ≥ ε γ n } .

It follows from (40) that

⟨ Y ˜ n ( t ) , θ ⟩ = ∫ ∣ x ∣ ∉ ( J n − 1 , J n ) ( 1 − cos ( t x ) ) 1 ∣ x ∣ D + I ⁄ 2 d m ( x ) , θ + ∫ ∣ x ∣ ∉ ( J n − 1 , J n ) sin ( t x ) 1 ∣ x ∣ D + I ⁄ 2 d m ′ ( x ) , θ = ∫ ∣ x ∣ ∉ ( J n − 1 , J n ) ( 1 − cos ( t x ) ) 1 ∣ x ∣ D * + I ⁄ 2 θ , d m ( x ) + ∫ ∣ x ∣ ∉ ( J n − 1 , J n ) sin ( t x ) 1 ∣ x ∣ D * + I ⁄ 2 θ , d m ′ ( x ) .

By Lemmas 2.1 and Facts (i)–(vii), we have

(41) γ n − 1 1 ∣ x ∣ D * + I ⁄ 2 θ = ‖ U * ( ( K 2 , 17 ( 1 − ε ) − 1 h n ⁄ L 2 h n ) D ) * θ ‖ − 1 ⋅ 1 ∣ x ∣ I ⁄ 2 1 ∣ x ∣ D * θ ≤ K ‖ U ‖ 1 ∣ x ∣ 1 ⁄ 2 ‖ ( ( h n ⁄ L 2 h n ) D ) * θ ‖ − 1 ⋅ 1 ∣ x ∣ D * θ ≤ K ‖ U ‖ 1 ∣ x ∣ 1 ⁄ 2 L 2 h n h n ∣ x ∣ 1 λ ( θ ) − ε if  ∣ x ∣ ≥ J n , K ‖ U ‖ 1 ∣ x ∣ 1 ⁄ 2 L 2 h n h n ∣ x ∣ 1 λ ( θ ) + ε if  ∣ x ∣ ≤ J n − 1 .

Thus,

(42) E [ ∣ γ n − 1 ⟨ Y ˜ n ( h n ) , θ ⟩ ∣ 2 ] = ∫ ∣ x ∣ ∉ ( J n − 1 , J n ) ( 1 − cos ( h n x ) ) γ n − 1 1 ∣ x ∣ D * + I ⁄ 2 θ 2 d x ≤ K ‖ U ‖ 2 h n − 2 λ ( θ ) − 2 ε ( L 2 h n ) 2 λ ( θ ) + 2 ε ∫ ∣ x ∣ ≤ J n − 1 ( 1 − cos ( h n x ) ) d x ∣ x ∣ 2 λ ( θ ) + 1 + 2 ε + K ‖ U ‖ 2 h n − 2 λ ( θ ) + 2 ε ( L 2 h n ) 2 λ ( θ ) − 2 ε ∫ ∣ x ∣ ≥ J n ( 1 − cos ( h n x ) ) d x ∣ x ∣ 2 λ ( θ ) + 1 − 2 ε ≤ K 3 , 8 ( h k J n − 1 ) 2 − 2 λ ( θ ) − 2 ε ( log n ) 2 λ ( θ ) + 2 ε + ( h n J n ) − 2 λ ( θ ) + 2 ε ( log n ) 2 λ ( θ ) − 2 ε .

To obtain the second inequality from bottom, in the first integral we bound 1 − cos ( t x ) by ∣ t ∣ 2 ∣ x ∣ 2 , and the second one by 2 to obtain the required bound. Thus, since λ ( θ ) ≥ a d − 1 > 1 ,

h k J n − 1 ≤ ( n − 1 ) e − q ( n − 1 ) q − 1 , h n J n = n ,

we have

(43) E [ ∣ γ n − 1 ⟨ Y ˜ n ( h n ) , θ ⟩ ∣ 2 ] ≤ K 3 , 9 n − 2 λ ( θ ) + ε

for all large n .

By (43) and Corollary 3.2 in [28], p.59, we obtain

(44) ∑ ∞ P ( G ˜ n ) ≤ K ∑ ∞ exp ( − K 3 , 10 ε 2 n 2 λ ( θ ) − ε ) < ∞ .

This implies that

(45) limsup n → ∞ γ n − 1 M ˜ ( h n ) ≤ ε a.s.

It follows from (20) that

(46) ∑ ∞ P ( M ( h n ) ≤ γ n ) ≥ K ∑ ∞ n − q ( 1 − ε ) = ∞

by choosing q > 1 small enough such that q ( 1 − ε ) < 1 , where the sums are over all n large enough to make γ n < 1 .

It follows easily that

P ( G ¯ n ) ≥ P ( G n ) − P ( G ˜ n ) ,

which, together with (44) and (46), yields

∑ ∞ P ( G ¯ n ) = ∞ .

Since Y n ( ⋅ ) , n = 1 , 2 , … are independent, by the Borel-Cantelli lemma, we obtain

(47) limsup n → ∞ γ n − 1 M ¯ n ≤ 1 − ε a.s.

From Lemma 2.1, we have f h n − 1 γ n − 1 ≤ K 3 , 11 . It follows from (45) and (47) that

liminf n → ∞ f h n M ( h n ) ≤ K 3 , 11 liminf n → ∞ γ n − 1 M ( h n ) ≤ K 3 , 11 ( liminf n → ∞ γ n − 1 M ¯ ( h n ) + limsup n → ∞ γ n − 1 M ˜ ( h n ) ) ≤ K 3 , 11 a.s.

This yields that (39) holds.

We have thus established that

K 3 , 6 ≤ liminf h → 0 + f h M ( h ) ≤ K 3 , 7 a.s.

Lemma 3.1 guarantees that the liminf is constant. The proof is completed.□

3.3 Functional laws of the iterated logarithm

We first recall some well-known facts about reproducing kernel Hilbert spaces [17]. Let E be a separable Banach space with dual E * , μ a centered Gaussian measure on E . Let π denote the canonical map of E * into L 2 ( E , μ ) , and let E μ * denote the closure in L 2 ( μ ) of π ( E * ) . For any k ∈ E μ * , the measure k ( x ) μ ( d x ) has a barycenter

ϕ ( k ) = ∫ E x k ( x ) μ ( d x ) ∈ E ,

where the integral may be interpreted either in the Pettis or the Bochner sense. The map k → ϕ ( k ) is linear and injective. The reproducing kernel Hilbert space (RKHS) H μ of μ is the range ϕ ( E μ * ) ⊂ E with the inner product

⟨ ϕ ( k ) , ϕ ( ξ ) ⟩ μ = ∫ E k ( x ) ξ ( x ) μ ( d x ) , k , ξ ∈ E μ * .

If we put ϕ ˆ = ϕ ∘ π , then ϕ ˆ ( E * ) is dense in H μ . We shall write ‖ f ‖ μ 2 = ⟨ f , f ⟩ μ for f ∈ H μ .

The following results are well known (see [29] or [30]).

Proposition 3.2

Let V be a convex, symmetric, measurable subset of E. For all f ∈ H μ and ξ ∈ E * ,

(48) μ ( f + V ) ≤ μ ( V ) exp − 1 2 ‖ f ‖ μ 2 + 1 2 ‖ f − ϕ ˆ ( ξ ) ‖ μ 2 + sup x ∈ V ξ ( x ) .

Furthermore,

(49) μ ( f + V ) ≥ μ ( V ) exp − 1 2 ‖ f ‖ μ 2 .

Proposition 3.3

Let V be a convex, symmetric, bounded, measurable subset of E of positive μ -measure. If f ∈ H μ , then

(50) lim t → ∞ t − 2 { log μ ( t f + V ) − log μ ( V ) } = − 1 2 ‖ f ‖ μ 2 .

Furthermore, the convergence is uniform over all such sets V of diameter less than 1, and the limit is a lower bound for all t.

Let X = { X ( t ) , t ∈ R } be an operator fractional Brownian motion in R p with exponent D satisfying that 0 < a 1 , a d < 1 and detCov ( X ( 1 ) ) > 0 . As in Section 1, for fixed h ∈ R + and θ ∈ Γ , let

ϒ h ( s ) = ⟨ h − D X ( h s ) , θ ⟩ 2 L 2 h , 0 ≤ s ≤ 1 ,

(51) g h = 1 φ ( 1 ⁄ L 2 h ) and ρ h = g h 2 L 2 h .

Let H D ⊆ C [ 0 , 1 ] be the RKHS of the kernel

R ( s , t ) = 1 2 ( φ 2 ( s ) + φ 2 ( t ) − φ 2 ( ∣ s − t ∣ ) ) , 0 ≤ s ≤ 1 , 0 ≤ t ≤ 1 .

H D is the RKHS corresponding to the centered Gaussian measure μ on the Banach space C [ 0 , 1 ] induced by { ⟨ X ( s ) , θ ⟩ , 0 ≤ s ≤ 1 } . As mentioned earlier, let ⟨ f , g ⟩ D be the inner product in H D and let ‖ f ‖ ∞ be the sup-norm on C [ 0 , 1 ] . If f ∈ H D , then ∣ f ( s ) − f ( t ) ∣ 2 ≤ φ 2 ( ∣ s − t ∣ ) ⟨ f , f ⟩ D .

We first take care of the case ⟨ f , f ⟩ D < 1 . The case ⟨ f , f ⟩ D = 1 is more delicate. Let C * denote the dual of the Banach space C [ 0 , 1 ] . Combining the analysis of [17] of fractional Brownian motion with our technique of treating operator fractional Brownian motion, we obtain the following result.

Theorem 3.1

(I) Let ⟨ f , f ⟩ D < 1 . Then

(52) liminf h → 0 + ρ h ‖ ϒ h − f ‖ ∞ = κ ( f ) a.s. ,

where κ ( f ) is a constant satisfying

2 − 1 ⁄ 2 K 2 , 5 K 2 , 18 1 λ ( θ ) ( 1 − ⟨ f , f ⟩ D ) − 1 λ ( θ ) ≤ κ ( f ) ≤ 2 − 1 ⁄ 2 K 2 , 6 K 2 , 17 1 λ ( θ ) ( 1 − ⟨ f , f ⟩ D ) − 1 λ ( θ )

with λ ( θ ) as in Section 2, K 2 , 5 and K 2 , 6 as in (10), K 2 , 17 and K 2 , 18 as in (20).

(II) Let ⟨ f , f ⟩ D = 1 . Then

(53) liminf h → 0 + ρ h ‖ ϒ h − f ‖ ∞ = ∞ a.s. ,

whereas

(54) liminf h → 0 + ρ ˆ h ‖ ϒ h − f ‖ ∞ < ∞ a.s. ,

where ρ ˆ h = ( L 2 h ) − ( 2 + λ ( θ ) ) 2 ( λ ( θ ) + λ 2 ( θ ) ) ρ h .

Theorem 3.2

(I) If ⟨ f , f ⟩ D = 1 and f ∈ ϕ ˆ ( C * ) , then

(55) liminf h → 0 + ρ ˆ h ‖ ϒ h − f ‖ ∞ > 0 a.s.

(II) If ⟨ f , f ⟩ D = 1 and f ∉ ϕ ˆ ( C * ) , then

(56) liminf h → 0 + ρ ˆ h ‖ ϒ h − f ‖ ∞ = 0 a.s.

In the proofs of both Theorems 3.1 and 3.2, we shall need the following lemma.

Lemma 3.2

For 0 < m ≤ h ≤ r < e − 1 and f ∈ H D ,

(57) ρ h ‖ ϒ h − f ‖ ∞ ≥ g r 2 L 2 m ‖ ϒ m − f ‖ ∞ − ( 2 L 2 m − 2 L 2 r ) ‖ f ‖ ∞ − sup 0 ≤ t ≤ m sup 0 ≤ s ≤ r − m ∣ ⟨ m − D ( X ( t + s ) − X ( t ) ) , θ ⟩ ∣ − 2 e ‖ D ‖ log ( r ⁄ m ) sup 0 ≤ t ≤ m sup 0 ≤ s ≤ r − m ‖ m − D ( X ( t + s ) − X ( t ) ) ‖ .

Proof

Note that

(58) ‖ ⟨ h − D X ( ( ⋅ ) h ) , θ ⟩ − 2 L 2 h f ‖ ∞ ≥ ‖ ⟨ m − D X ( ( ⋅ ) m ) , θ ⟩ − 2 L 2 m f ‖ ∞ − ( 2 L 2 m − 2 L 2 h ) ‖ f ‖ ∞ − ‖ ⟨ h − D X ( ( ⋅ ) h ) , θ ⟩ − ⟨ m − D X ( ( ⋅ ) m ) , θ ⟩ ‖ ∞ .

The last term of the aforementioned inequality is less than

(59) ‖ ⟨ h − D X ( ( ⋅ ) h ) , θ ⟩ − ⟨ h − D X ( ( ⋅ ) m ) , θ ⟩ ‖ ∞ + ‖ ⟨ h − D X ( ( ⋅ ) m ) , θ ⟩ − ⟨ m − D X ( ( ⋅ ) m ) , θ ⟩ ‖ ∞ = ‖ ⟨ m − D ( X ( ( ⋅ ) h ) − X ( ( ⋅ ) m ) ) , ( h ⁄ m ) − D * θ ⟩ ‖ ∞ + ‖ ⟨ m − D ( X ( ( ⋅ ) h ) − X ( ( ⋅ ) m ) ) , ( ( h ⁄ m ) − D * − I ) θ ⟩ ‖ ∞ ≤ ‖ ⟨ m − D ( X ( ( ⋅ ) h ) − X ( ( ⋅ ) m ) ) , θ ⟩ ‖ ∞ + 2 ‖ ⟨ m − D ( X ( ( ⋅ ) h ) − X ( ( ⋅ ) m ) ) , ( ( h ⁄ m ) − D * − I ) θ ⟩ ‖ ∞ .

The first term in the last inequality in (59) is less than

(60) sup 0 ≤ t ≤ m sup 0 ≤ s ≤ r − m ∣ ⟨ m − D ( X ( t + s ) − X ( t ) ) , θ ⟩ ∣ .

Since ‖ t D − I ‖ = ‖ e D log t − I ‖ ≤ e ‖ D ‖ log t for 1 < t < 2 , the second term in the last inequality in (59) is less than

(61) e ‖ D ‖ log ( r ⁄ m ) sup 0 ≤ t ≤ m sup 0 ≤ s ≤ r − m ‖ m − D ( X ( t + s ) − X ( t ) ) ‖ .

Combining (58)–(61), we obtain (57).□

Proof of Theorem 3.1

By Propositions 3.3 and 2.1, and Lemmas 2.1 and 3.2, following the same lines as the proof Theorem 4.3 of [17], we obtain (52).

We now show (53). Let h n = exp ( − n ( log n ) − 1 ) . For any large constant Q > 0 , any τ > 0 and n > n 0 ( δ ) ,

( L 2 h n ) − 1 log P ( ‖ ⟨ h n − D X ( ( ⋅ ) h n ) , θ ⟩ − 2 L 2 h n f ‖ ∞ ≤ φ ( Q ⁄ L 2 h n ) ) ≤ − K 3 , 12 Q − 1 − ⟨ f , f ⟩ D + τ .

Since ⟨ f , f ⟩ D = 1 , choosing τ < K 3 , 12 Q − 1 , it follows that the sum of above probabilities is finite. By using Lemmas 2.1 and 3.2, we can conclude that

(62) liminf h → 0 + ρ h ‖ ϒ h − f ‖ ∞ ≥ Q 1 2 λ ( θ ) a.s.

Letting Q ↑ ∞ , we obtain (53). In our proof of (54), we follow Theorem 4.4 of [17]. For n = 2 , 3 , … , let h n = n − n . Put

γ n = Q ρ ˆ h n − 1 , f n = ( 1 − γ n ‖ f ‖ ∞ − 1 ) f

for a suitable, large constant Q , and 0 < τ < 1 . It follows from Proposition 3.2 that

(63) P ( ‖ ϒ h n − f ‖ ∞ ≤ 2 γ n ) ≥ P ( ‖ ϒ h n − f n ‖ ∞ ≤ γ n ) ≥ P ( ‖ ⟨ h n − D X ( ( ⋅ ) h n ) , θ ⟩ − 2 L 2 h n f n ‖ ∞ ≤ γ n 2 L 2 h n ) ≥ exp { − ⟨ f n , f n ⟩ D L 2 h n } P ( M ( 1 ) ≤ γ n 2 L 2 h n ) .

It follows from Lemma 2.1 that for any small ε > 0 ,

(64) K 3 , 13 h 1 λ ( θ ) − ε ≤ φ ( h ) ≤ K 3 , 14 h 1 λ ( θ ) + ε if  h ≥ 1 ,

and

(65) K 3 , 15 h 1 λ ( θ ) + ε ≤ φ ( h ) ≤ K 3 , 16 h 1 λ ( θ ) − ε if  h < 1 .

Since ⟨ f , f ⟩ D = 1 , it follows from Proposition 2.1 that for large n and small τ > 0 ,

(66) log P ( ‖ ϒ h n − f ‖ ∞ ≤ 2 γ n ) ≥ P ( ‖ ϒ h n − f n ‖ ∞ ≤ γ n 2 L 2 h n ) ≥ − ( 1 − γ n ‖ f ‖ ∞ − 1 ) 2 L 2 h n − K 3 , 17 γ n − λ ( θ ) − τ ( L 2 h n ) − λ ( θ ) 2 + τ ≥ − L 2 h n + ( 2 Q ‖ f ‖ ∞ − 1 − K 3 , 17 Q − λ ( θ ) − τ 2 − λ ( θ ) 2 + τ ) ( L 2 h n ) λ ( θ ) 2 ( 1 + λ ( θ ) ) − τ − Q 2 ‖ f ‖ ∞ − 2 ( L 2 h n ) − 1 1 + λ ( θ ) + τ .

Choosing Q large enough, (66) yields that for n > n 0 ( K 3 , 17 ) ,

P ( ‖ ϒ h n − f ‖ ∞ ≤ 2 γ n ) ≥ exp ( − L 2 h n ) = ( n log n ) − 1 .

This implies that

∑ P ( ‖ ϒ h n − f ‖ ∞ ≤ 2 γ n ) = ∞ .

Arguing as in the first half of the proof of Theorem 3.1(I), we have

(67) liminf n → ∞ ρ ˆ h n ‖ ϒ h n − f ‖ ∞ ≤ 2 Q a.s.

This proves (54).□

Proof of Theorem 3.2

We first prove (55). Assume that ⟨ f , f ⟩ D = 1 and that f = ϕ ˆ ( ξ ) for some ξ ∈ C * . Again we follow Theorem 4.4 of [17]. We shall use the notation

‖ g ‖ = sup { g ( ξ ) : ξ ∈ C [ 0 , 1 ] , ‖ ξ ‖ ∞ ≤ 1 } .

Let h n = exp ( − n ( log n ) − 1 − λ ( θ ) ) for n ≥ 2 , and put γ n = ε ρ ˆ h n − 1 for a suitable, small constant ε > 0 . By Proposition 3.2, we have for large enough n ,

(68) P ( ‖ ϒ h n − f ‖ ∞ ≤ γ n ) ≤ exp ( − L 2 h n + 2 L 2 h n ‖ g ‖ γ n ) P ( M ( 1 ) ≤ 2 L 2 h n γ n ) .

By (64), (65), and Proposition 2.1, (68) becomes, for small τ > 0 ,

P ( ‖ ϒ h n − f ‖ ∞ ≤ γ n ) ≤ exp ( − L 2 h n − ( K 3 , 18 ε − λ ( θ ) + τ 2 − λ ( θ ) 2 − τ − 2 ε ‖ g ‖ ) ( L 2 h n ) λ ( θ ) 2 ( 1 + λ ( θ ) ) − τ ) .

If ε is chosen small enough, then we have for n > n 0 ( ε ) ,

P ( ‖ ϒ h n − f ‖ ∞ ≤ γ n ) ≤ exp { − log n + ( 1 + λ ( θ ) ) log log n − K 3 , 19 ( log n ) λ ( θ ) 2 ( 1 + λ ( θ ) ) − τ } .

This yields that ∑ ∞ P ( ‖ ϒ h n − f ‖ ∞ ≤ γ n ) < ∞ . Hence,

(69) liminf n → ∞ ρ h n ‖ ϒ h n − f ‖ ∞ ≥ ε a.s.

This, together with Lemma 3.2 that

(70) liminf h → 0 + ρ h ‖ ϒ h − f ‖ ∞ ≥ ε a.s.

This completes the proof of (55).

We next show (56). Let h n = n − n . Given ε > 0 define γ n = ε ρ ˆ h n − 1 as mentioned earlier. Defined for f ∈ H D and r ≥ 0 by

I ( f , r ) = inf { ⟨ g , g ⟩ D : g ∈ H D , ‖ f − g ‖ ∞ ≤ r } .

From Lemma 1 of [31] or Theorem 4.4 of [17], we have that there is a unique element f n ∈ H D such that ‖ f − f n ‖ ∞ = γ n , and I ( f , γ n ) = ⟨ f n , f n ⟩ D . It follows from Proposition 3.3.1 that, for large n ,

(71) P ( ‖ ϒ h n − f ‖ ∞ ≤ γ n ) ≥ P ( ‖ ϒ h n − f n ‖ ∞ ≤ γ n ) ≥ exp { − ⟨ f n , f n ⟩ D L 2 h n } P ( M ( 1 ) ≤ γ n 2 L 2 h n ) ≥ exp { − I ( f , γ n ) L 2 h n − K 3 , 20 γ n − λ ( θ ) − τ ( 2 L 2 H n ) − λ ( θ ) 2 + τ } .

From Theorem 4.4 of [17], we have that if ⟨ f , f ⟩ D = 1 and f ∉ ϕ ˆ ( C * ) , then lim r ↓ 0 ( 1 − I ( f , r ) ) ⁄ r = ∞ . (71) becomes that for any large constant Q , small τ > 0 and all n > n 0 ( Q , τ ) ,

(72) P ( ‖ ϒ h n − f ‖ ∞ ≤ 2 γ n ) ≥ exp − ( 1 − Q γ n ) L 2 h n − K 3 , 21 γ n − λ ( θ ) − τ ( 2 L 2 H n ) − λ ( θ ) 2 + τ ≥ exp − L 2 h n + ( Q ε − K 3 , 22 ε − λ ( θ ) − τ 2 − λ ( θ ) 2 + τ ) ( L 2 h n ) λ ( θ ) 2 ( 1 + λ ( θ ) ) − τ .

By choosing Q > K 3 , 22 ε − 1 − λ ( θ ) − τ 2 − λ ( θ ) 2 + τ , we conclude that ∑ ∞ P ( ‖ ϒ h n − f ‖ ∞ ≤ 2 γ n ) = ∞ . Similar to the second half of the proof of (52), we obtain

(73) liminf n → ∞ ρ h n ‖ ϒ h n − f ‖ ∞ ≤ 2 ε a.s.

Letting ε ↓ 0 , we obtain (56).□

4 Conclusions

By applying techniques developed in [17], in this article, we obtain the estimations of small ball probability for spatial surfaces of operator fractional Brownian motion, including multivariate fractional Brownian motion. We obtain the strongly locally nondeterministic for operator fractional Brownian motion in any given direction θ . By applying these estimates, we obtain Chung type and functional laws of the iterated logarithm for operator fractional Brownian motion in any given direction θ . By combining our results and the Jordan decomposition theorem applied to the exponent D , it is possible to analyze Hölder regularities of these spatial surfaces by the real parts of the eigenvalues of the exponent D and the covariance matrix at the time point 1.

Looking forward, several avenues for future research are apparent. One is to extend these methods and results to more general classes of multivariate Gaussian random fields with different types of scaling laws. Further investigation into the applications of these findings in various applied fields, such as environmental statistics, financial mathematics, and engineering, could yield valuable insights and practical tools. In addition, exploring the implications of these Hölder regularities on the numerical simulation of spatial processes presents another rich area for continued study.

Acknowledgements

The authors wish to express their deep gratitude to referees for their valuable comments on an earlier version which improve the quality of this article.

  1. Funding information: The first author was supported by Humanities and Social Sciences of Ministry of Education Planning Fund of China (Grant No. 21YJA910005) and National Natural Science Foundation of China (Grant No. 11671115).

  2. Author contributions: The authors contributed equally to this work.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

References

[1] K. Sato, Self-similar processes with independent increments, Probab. Theory Related Fields 89 (1991), 285–300. 10.1007/BF01198788Search in Google Scholar

[2] R. G. Laha and V. K. Rohatgi, Operator self-similar stochastic processes in Rd, Stochastic Process. Appl. 12 (1981), 73–84. 10.1016/0304-4149(81)90012-0Search in Google Scholar

[3] W. N. Hudson and J. D. Mason, Operator-self-similar processes in a finite-dimensional space, Trans. Amer. Math. Soc. 273 (1982), 281–297. 10.1090/S0002-9947-1982-0664042-7Search in Google Scholar

[4] M. Maejima and J. D. Mason, Operator-self-similar stable processes, Stochastic Process. Appl. 54 (1994), 139–163. 10.1016/0304-4149(94)00010-7Search in Google Scholar

[5] G. Didier and V. Pipiras, Integral representations of operator fractional Brownian motion, Bernoulli 17 (2011), 1–33. 10.3150/10-BEJ259Search in Google Scholar

[6] F. Lavancier, A. Philippe, and D. Surgailis, Covariance function of vector self-similar process, Stat. Probab. Lett. 79 (2009), 2415–2421. 10.1016/j.spl.2009.08.015Search in Google Scholar

[7] F. Lavancier, A. Philippe, and D. Surgailis, A two-sample test for comparison of long memory parameters, J. Multivariate Anal. 101 (2010), 2118–2136. 10.1016/j.jmva.2010.04.003Search in Google Scholar

[8] J.-F. Coeurjolly, P. O. Amblard, and S. Achard, Wavelet analysis of the multivariate fractional Brownian motion, ESAIM Probab. Stat. 17 (2013), 592–604. 10.1051/ps/2012011Search in Google Scholar

[9] Y. Li, M. S. Pakkanen, and A. E. D. Veraart, Limit theorems for the realised semicovariances of multivariate Brownian semistationary processes, Stochastic Process. Appl. 155 (2023), 202–231. 10.1016/j.spa.2022.10.001Search in Google Scholar

[10] P.-O. Amblard, J.-F. Coeurjolly, F. Lavancier, and A. Philippe, Basic properties of the multivariate fractional Brownian motion, 2010, arXiv preprint, arXiv:1007.0828. Search in Google Scholar

[11] S. Kechagias and V. Pipiras, Definitions and representations of multivariate long-rangle dependent time serires, J. Time Ser. Anal. 36 (2015), 1–25. 10.1111/jtsa.12086Search in Google Scholar

[12] D. Debangan, D. Abhirup, and B. Sudipto, Graphical Gaussian process models for highly multivariate spatial data, Biometrika 109 (2021), 993–1014. 10.1093/biomet/asab061Search in Google Scholar PubMed PubMed Central

[13] T. Konstantopoulos and S. J. Lin, Fractional Brownian approximations of queueing networks, in: Stochastic Networks, Lecture Notes in Statistics, vol. 117, Springer, New York, 1996, pp. 257–273. 10.1007/978-1-4612-4062-4_13Search in Google Scholar

[14] R. Delgado, A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic, Stochastic Process. Appl. 117 (2007), 188–201. 10.1016/j.spa.2006.07.001Search in Google Scholar

[15] J. D. Mason and Y. Xiao, Sample path properties of operator-self-similiar Gaussian random fields, Theory Probab. Appl. 46 (2002), 58–78. 10.1137/S0040585X97978749Search in Google Scholar

[16] W. Wang, The moduli of continuity for operator fractional Brownian motion, J. Th. Probab. (2023), DOI: https://doi.org/10.1007/s10959-023-01307-z. 10.1007/s10959-023-01307-zSearch in Google Scholar

[17] D. Monrad and H. Rootzén, Small values of Gaussian processes and functional laws of the iterated logarithm, Probab. Th. Rel. Fields 101 (1995), 173–192. 10.1007/BF01375823Search in Google Scholar

[18] M. Csörgő and P. Révész, Strong Approxiamtions in Probability and Statistics, Academic Press, New York, 1981. Search in Google Scholar

[19] Q.-M. Shao, Bounds and estimators of a basic constant in extreme value theory of Gaussian processes, Statist Sinica 7 (1996), 245–257. Search in Google Scholar

[20] W. V. Li and W. Linde, Existence of small ball constants for fractional Brownian motions, C. R. Acad. Sci. Paris 326 (1998), 1329–1334. 10.1016/S0764-4442(98)80189-4Search in Google Scholar

[21] W. Wang and Y. Xiao, The Csörgő-Révész moduli of non-differentiability of fractional Brownian motion, Stat. Probab. Lett. 150 (2019), 81–90. 10.1016/j.spl.2019.02.016Search in Google Scholar

[22] M. M. Meerschaert and H. P. Scheffler, Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice, Wiley Interscience, New York, 2001. Search in Google Scholar

[23] M. W. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra, in: Pure and Applied Mathematics, vol. 60, Academic Press, New York, London, 1974. Search in Google Scholar

[24] W. Wang, Invariance principles for generalized domains of semistable attraction, Stochastic Process. Appl. 124 (2014), 1–17. 10.1016/j.spa.2013.07.004Search in Google Scholar

[25] M. Talagrand, New Gaussian estimates for enlarged balls, Geo. Func. Anal. 3 (1993), 502–526. 10.1007/BF01896240Search in Google Scholar

[26] W. V. Li and Q.-M. Shao, Gaussian processes: inequalities, small ball probabilities and applications, in: C. R. Rao and D. Shanbhag (Eds.), Stochastic Processes: Theory and Methods, Handbook of Statistics, vol. 19, North-Holland, 2001, pp. 533–597. 10.1016/S0169-7161(01)19019-XSearch in Google Scholar

[27] M. M. Meerschaert, W. Wang, and Y. Xiao, Fernique type inequality and moduli of continuity for anisotropic Gaussian random fields, Tran. Amer. Math. Soc. 365 (2013), 1081–1107. 10.1090/S0002-9947-2012-05678-9Search in Google Scholar PubMed PubMed Central

[28] M. Ledoux and M. Talagrand, Probabilities in Banach Spaces, Springer-Verlag, New York, 1991. 10.1007/978-3-642-20212-4Search in Google Scholar

[29] C. Borell, A note on Gauss measures which agree on small balls, Ann. Inst. Henri Poincaré Probab. Stat. 3 (1977), 231–238. Search in Google Scholar

[30] A. deAcosta, Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm, Ann. Probab. 11 (1983), 78–101. 10.1214/aop/1176993661Search in Google Scholar

[31] K. Grill, A lim inf result in Strassenas law of the iterated logarithm, Probab. Theory Related Fields 89 (1991), 149–157. 10.1007/BF01366903Search in Google Scholar
Received: 2023-09-10
Revised: 2024-07-17
Accepted: 2024-07-24
Published Online: 2024-08-21

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0045
Keywords for this article
operator self-similarity; operator fractional Brownian motion; small ball probability; Chung type laws of the iterated logarithm; strong local nondeterminism
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 14.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2024-0045/html
Scroll to top button