Strong limit theorems for anisotropic self-similar fields

Our paper starts from presentation and comparison of three definitions for the selfsimilar field. The interconnection between these definitions has been established. Then we consider the Lamperti scaling transformation for the self-similar field and investigate the connection between the scaling transformation for such field and the shift transformation for the corresponding stationary field. It was also shown that the fractional Brownian sheet has the ergodic scaling transformation. The strong limit theorems for the anisotropic growth of the sample paths of the self-similar field at 0 and at ∞ for the upper and lower functions have been proved. It was obtained the upper bound for growth of the field with ergodic scaling transformation for slowly varying functions. We present some examples of iterated log-type limits for the Gaussian self-similar random fields.


Introduction
A self-similar process is a process invariant by distribution under specific time and/or space scaling.Namely, a stochastic process {(),  ∈ ℝ} is self-similar with index  ≥ 0, if for any  > 0 {(),  ∈ ℝ} = { (),  ∈ ℝ}, where = denotes the equality of the finite-dimensional distributions.The books by Embrechts & Maejima [6] and Samorodnitsky & Taqqu [14] are devoted to the theory of self-similar processes.
A classical example of a self-similar process with index  ∈ (0, 1) is a fractional Brownian motion { (),  ∈ ℝ + } (ℝ + = [0, +∞)) with a corresponding Hurst index.This process has centered stationary increments and the following covariance function The investigation of self-similar random fields (multiparameter processes) was caused by the evidence of the self-similarity property of phenomena in climatology, environmental sciences, etc. (see [10,13]).In particular, so called anisotropic random fields are used for modeling phenomena in spatial statistics, statistical hydrology and image processing (see [2,3,5]).The attempts to extend the self-similarity concept from processes to fields resulted in arising of the several approaches.In our paper we present three different definitions of the self-similar fields and establish the interconnection between them.We show that the covariance function of the centered Gaussian field determines the self-similarity property and its type.The definitions of the fractional Brownian fields and sheets are also presented in the paper.It is proved that they are self-similar fields but according to the different definitions.We consider the fields which are self-similar with respect to every coordinate with individual index.Such fields are used to call anisotropic and in the Brownian case they usually are called as Brownian sheets.The paper's aim is to investigate the asymptotic growth of the sample paths of these fields.
We introduce the notions of upper and lower functions for the sample paths of the random field which are similar to the paper [15] and prove the zero-one law for such functions in the case of growth at 0 and at ∞.We also assume the ergodicity of the scaling transformation.The ergodicity property should be proved independently for every particular case and this can be easily done for the stationary fields and processes.
Let us also mention that the non-singular self-similar process has not to be stationary.But there is a one-to-one correspondence between self-similar and stationary processes.For every self-similar process  with index  > 0, its Lamperti transformation  = {() =  − ( )} is a stationary process.The Lamperti transformation for anisotropic random fields was introduced in the paper [7] and there was established the correspondence between self-similar and stationary random fields as well.
In the present paper it is proved that the ergodicity of the shift transformation for the corresponding stationary field is a sufficient condition for the ergodicity of the scaling transformation.For the Gaussian fields the last statement can be ensured by the proper conditions on the covariance function.In particular, we prove that the fractional Brownian sheet has the ergodic scaling transformation.
In this paper the strong limit theorems for the anisotropic growth of the sample paths of the self-similar fields for the upper and lower functions arising in the zeroone law is proved.The similar theorems for the self-similar stochastic processes were proved in the paper [8].Application of these theorems to the Gaussian fields allows to obtain the iterated log-type lows.Comparing the results for the fractional Brownian fields and sheets with the results from the paper [11] we can conclude that our theorems enable us to obtain more precise estimates.
The paper is organized as follows.Section 2 contains the different definitions of a self-similar field and the interconnection between them is proved.We focused on the Gaussian case and present the definitions of the fractional Brownian field and sheet.In Section 3, the Lamperti transformation for the self-similar field is considered and the connection between the scaling transformation for such a field and the shift transformation for the corresponding stationary field is stated proved.It is shown that the fractional Brownian sheet has the ergodic scaling transformation.In Section 4, we introduce the definitions of the upper and lower functions for asymptotic growth of the sample paths of the self-similar field.The zero-one law is proved for the fields with ergodic scaling transformation.The strong limit theorems for asymptotic growth of sample paths at 0 and at ∞ are obtained in Section 5. We also establish there the upper bounds for growth of the field with ergodic scaling transformation for the case of the slowly varying functions.In Section 6 we apply the theorem to prove the iterated log-type laws for the Gaussian self-similar fields.

Definition of self-similarity for random fields
Let us start from giving three different definitions of the self-similar random fields and then show their interrelation.We assume that {Ω, ℱ, } is a standard probability space defining all the random objects considered further on.
Hereinafter we shall use the designation  ⋅  in order to denote the vector consisting of the coordinatewise products of two vectors ,  ∈ ℝ  ⋅  = ( 1  1 , … ,   ), In addition it is possible to give the third definition of the self-similar field as a field which is self-similar with respect to every time coordinate.For arbitrary  > 0 and 1 ≤  ≤  we put  1 = 1, … ,  −1 = 1,  = ,  +1 = 1, … ,  = 1,  = ( 1 , … ,  ).Then X is self-similar with respect to the -th coordinate.
The lemma is proved.
Then the field  1 is self-similar with index  = ( 1 , … ,  ) by Definition 2.2, and the field  2 is self-similar with index  by Definition 2.1.
Proof.The fact that the finite-dimensional distributions of the centered Gaussian fields are uniquely determined by the covariance function implies the lemma's proof.Within the lemma's conditions the covariance matrices Σ 1 and Σ 2 of the corresponding random fields  1 and  2 have the following properties: The lemma is proved.
Let us give a few examples of Gaussian self-similar random fields.
Definition 2.4.As a standard Levy fractional Brownian field with index  > 0 we shall call a centered Gaussian random field  = { (),  ∈ ℝ + } with a covariance function where ‖⋅‖ is set for the Euclidean norm in ℝ .This field is self-similar by Definition 2.1 (see [14], Example 8.1.3).
The Lévy fractional Brownian field is isotropic.This field is the only one in law Gaussian self-similar field within Definition 2.1 with stationary isotropic increments.Definition 2.5.As a standard fractional Brownian sheet with index  = ( 1 , … ,  ), 0 <  < 1,  = 1,  we shall call a centered Gaussian random field   = {  (),  ∈ ℝ + } with a covariance function This field is self-similar by Definition 2.2 and has stationary rectangular increments.The proof of this property for the ℝ 2 case can be found in the paper [1].A similar property for the case  > 2 can be easily proved as well.
Remark 2.1.A random field satisfying Definition 2.1 is not necessary self-similar in a sense of Definition 2.2.Indeed, let us consider the Levy fractional Brownian field { (),  ∈ ℝ + }.It is self-similar by Definition 2.1 ([14], Example 8.1.3).We intend to prove that this field is not self-similar by Definition 2.2.
But, if the field satisfies Definition 2.2, then there should be

Self-similar fields with ergodic scaling transformation
Further in the paper, we assume that the fields satisfy Definition 2.2 and are real-valued and continuous in probability.Under such assumptions we could work with separable versions without loss of generality.Moreover, we shall consider only the case  = 2 since switching to the parameter of the higher dimension is rather technical.
+ } has the same finite-dimensional distributions as the field .
We shall call the field  self-similar with ergodic scaling transformation if   is ergodic.Further in this section we shall assume that any scaling transformation   ,  ∈ (0, +∞) 2 ,  ≠ (1, 1) for the self-similar field is ergodic.
It follows from the interconnection between the transformations   and   that the ergodicity of the field  =    implies the ergodicity of the field .In particular, the ergodicity of the scaling transformation for the Gaussian stationary processes follows from the covariance function properties.The class of the stationary fields that are ergodic is quite wide.Let us show that fractional Brownian sheet is ergodic.
+ } be fractional Brownian sheet with index  = ( 1 ,  2 ) ∈ (0, 1) 2 (Definition 2.5).Then   is the self-similar field with ergodic scaling transformation and the stationary field  =     is centered with the following covariance function Proof.The proof of the equality (2) can be found in the paper [7].All we need to do is to prove that the field  is ergodic.Let's check the conditions of Theorem 3.1.The covariance function  is continuous and bounded, and can be represented as follows: Thus, it follows from Theorem 3.1 that the field  is stationary with ergodic shift transformation.This implies that the corresponding anisotropic Brownian sheet   is self-similar with ergodic scaling transformation.The corollary is proved.

Upper and lower functions for ergodic fields
In this section we continue considering of the self-similar fields with ergodic scaling transformation and prove the zero-one laws for the asymptotic growth of the field's sample paths.Let us introduce the following definitions.
So, the following events occur with probability one This means that  0 , ≤  0 , ≤  0 , +  a.s.Since  > 0 is arbitrary, it concludes the corollary statement.The proof for the case of the growth at ∞ can be done in a similar way.
If we consider slowly varying functions we are able to obtain more specific result for the functionals  0 Λ, and  ∞ Λ, .

Proof. Let us introduce an auxiliary function
The function  is not necessary monotone in every coordinate.But we can show that  is monotone on some neighborhood of 0, if  is slowly varying at 0 and ∞ if  is slowly varying at ∞.
Let us consider the case when  1 >  1 ,  2 <  2 and investigate the growth at 0. We intend to prove that the function  increases in the first coordinate and decreases in the second one on some neighborhood of 0. It follows from the equality (3) that for any  1 < 1,  2 > 1 the following holds true Then, for 0 <  1 ∨  2 <  we get Therefore, for  < ( 1 − 1 there exists such  > 0 that the following inequality is true ( ⋅ ) ≤ (), ∀ 1 ∨  2 < .

Strong limit theorems
This section is devoted to strong limit theorems for real-valued self-similar fields within Definition 2.2.Let us prove these theorems for the function  1  1  2 2 (),  ∈ ℝ 2 + , arising in Theorem 4.1, for the fields with ergodic scaling transformation.It worth to mention that it is possible to prove the theorems in this section without imposing the additional condition about ergodicity of the scaling transformation.
We use the following notation defined for the self-similar field  = {(),  ∈ ℝ 2 + } with index  ∈ (0, +∞) 2 : Since the distributions of a self-similar field are invariant under the scale transformation   , all distribution properties can be concentrated on any finite interval.That is why all theorems within this section deal with the random variable  * defined by the values of the random field on the unit square.The following theorems are focused on establishing the sufficient conditions for the function to be upper or lower for the self-similar field.Let's start from proving one auxiliary result.
Proof.It follows from the self-similarity of the field that and therefore

𝐏( sup
Since  is positive and non-decreasing function, the Chebyshev's inequality implies that
Further in the text we shall use the following notation  = (1, 1).
Taking into account the inequality (7) we obtain
According to the integral criterion of series convergence for the positive non-decreasing function  (( )),  > 0 it can be concluded that the series converges if Thus, the integral ( , ) is finite by the condition () of the theorem.
For the case  1 ≥  2 we get the same inequality using the similar reasoning.So, a.s., and lim sup The theorem is proved.
The last series converges if Let's make the substitution  =  − in the integral ( , ).Then  = −/( ln ) and The integral ( , ) is finite by the condition () of the theorem.Thus, it follows from the Borel-Cantelli's lemma that there exists with probability one such a number  0 () that ( ) = 0 for all  ≥  0 ().It means that for all  > 1 and  ≥  0 () sup Furthermore, for every  > 1,  > 1 and  >  0 () we choose the point  = ( 1 ,  2 ) in such a way that  − −1 ≤  1 ≤  − and  − − −1 ≤  2 ≤  − − .Then, we obtain with probability one the following and lim sup The theorem is proved.
Now we can use these theorems for the self-similar fields with ergodic scaling transformation.The following corollary gives the sufficient conditions for the function to be upper one for such fields.Then for any  > 0 and an arbitrary slowly varying function  ∶ ℝ 2 + → (0, +∞) with respect to the growth at 0 (at ∞)  0 − , = 0 a.s.
Now we consider the behavior on ∞.Let us check the condition () of Theorem 5.1 for the function  ∞ : The condition () is also fulfilled since Thus, Theorem 5.1 implies that  ∞ + ,1 ≤ 1 a.s.Since the constant 1 can be regarded as slowly varying function, so it follows from Lemma 4.1 that  ∞ + ,1 = 0 or ∞ a.s.Therefore,  ∞ + ,1 = 0 a.s.for any  > 0. Let us now consider the behavior at 0. We check the condition () of Theorem 5.2 for the function  0 .
Proof.Using the similar argumentation as in Lemma 5.1 we obtain ( sup ).
Since the condition of the lemma implies that the function  is positively defined and continuous, then )( ).
The lemma is proved.
Theorem 5.3.Let  ∶ ℝ + → (0, +∞) be such a continuous non-increasing function that [( * )] is finite.We assume that a function  ∶ ℝ 2 + → (0, +∞) is continuous and satisfies the conditions: Proof.Let  > 1, ,  ∈ ℕ.We put  =  1  ( + ) 2 ( ,  + ) and define a sequence of the random events The following inequality for the probability of such events follows from Lemma 5.2 In order to prove the theorem we shall show that starting from some number the events  have zero probability.For this we need to prove the convergence of the series ∑ ∞ =1 ( ).
Let us recall that the functions  and  are non-decreasing under the theorem conditions and their superposition ((⋅, ⋅)) is a non-decreasing function in every coordinate.So, the reasons similar to the ones from Theorem 5.1 will lead us to the relations .
Since the function (( ,  )),  > 0 is non-decreasing, the integral criterion of series convergence implies that it is sufficient to prove the finiteness of the integral (, ) = ∫ So, the integral (, ) is finite according to the condition () of the theorem.Therefore, the series ∑ ∞ =1 ( ) is convergent.It follows from the Borel-Cantelli's lemma that there exists with probability one such a number  0 () that for all  ≥  0 () ∶ ( ) = 0.It means that Now, for arbitrary  > 1,  > 0 and  >  0 () we choose such a point  = ( 1 ,  2 ) that  ≤  1 ≤  +1 ,  + ≤  2 ≤  + +1 .Then, the following is true with probability one And therefore, The theorem is proved.

Strong limit theorems for Gaussian fields
Let us consider a few examples of how Theorems 5.1 and 5.3 can be applied to centered Gaussian fields.In this section we assume that the real-valued Gaussian fields have continuous sample paths.
The first condition of Theorems 5.1 and 5.3 is an existence of such a non-decreasing function  and a non-increasing function  that [ ( * )] < +∞, [( * )] < +∞.It is not so easy to check these conditions directly.But there are a lot of well-known results for the Gaussian fields concerning the tail probability behavior and the probability of the small deviations.The following lemma shows how this information can be utilized for checking the first condition of Theorems 5.1 and 5.3.Lemma 6.1.Let  ,  ∶ ℝ + → (0, +∞),  be non-decreasing,  be non-increasing,  ,  ∈  1 (ℝ + ).We assume that  is a positive random variable and the functions ,  ∶ ℝ + → ℝ + are such that ( > ) ≤ () and ( ≤ ) ≤ ().If The lemma is proved.