A Class of Negatively Fractal Dimensional Gaussian Random Functions

Let x t be a locally self-similar Gaussian random function. Denote by rxx τ the autocorrelation function ACF of x t . For x t that is sufficiently smooth on 0,∞ , there is an asymptotic expression given by rxx 0 − rxx τ ∼ c|τ | for |τ | → 0, where c is a constant and α is the fractal index of x t . If the above is true, the fractal dimension of x t , denoted by D, is given by D D α 2 − α/2. Conventionally, α is strictly restricted to 0 < α ≤ 2 so as to make sure that D ∈ 1, 2 . The generalized Cauchy GC process is an instance of this type of random functions. Another instance is fractional Brownian motion fBm and its increment process, that is, fractional Gaussian noise fGn , which strictly follow the case ofD ∈ 1, 2 or 0 < α ≤ 2. In this paper, I claim that the fractal index α of x t may be relaxed to the range α > 0 as long as its ACF keeps valid for α > 0. With this claim, I extend the GC process to allow α > 0 and call this extension, for simplicity, the extended GC EGC for short process. I will address that there are dimensions 0 ≤ D α < 1 for 2 < α ≤ 4 and further D α < 0 for 4 < α for the EGC processes. I will explain that x t with 1 ≤ D < 2 is locally rougher than that with 0 ≤ D < 1. Moreover, x t withD < 0 is locally smoother than that with 0 ≤ D < 1. The local smoothest x t occurs in the limit D → −∞. The focus of this paper is on the fractal dimensions of random functions. The EGC processes presented in this paper can be either long-range dependent LRD or short-range dependent SRD . Though applications of such class of random functions forD < 1 remain unknown, I will demonstrate the realizations of the EGC processes for D < 1. The above result regarding negatively fractal dimension on random functions can be further extended to describe a class of random fields with negative dimensions, which are also briefed in this paper.


Introduction
Conventionally, for a time series, or a random function x t , such as fully developed ocean wave series, we need not discuss its fractal dimension, as can be judged from the power spectra discussed by Massel 1 , the Specialist Committee on Waves of the 23rd ITTC 2 , The literature with respect to the time series with D ∈ 1, 2 is rich; see, for example, 4-68 , simply a drop in the bucket in the field.However, how to represent D ∈ 0, 1 and D < 0 in particular for time series remains an open problem.This paper gives a solution to this problem in the case that x t is a locally self-similar Gaussian random function the ACF of which follows the form of the GC process.
There are various definitions of dimensions Mandelbrot 15 , such as the Minkowski dimension, the Rényi dimension, the Hausdorff dimension, the packing dimension, the boxcounting dimension, the correlation dimension.Those dimensions may not be equal for a specific object but this does not matter.What an important thing is whether there are objects the dimensions of which are negative and how to represent negative dimensions of objects Mandelbrot 69 .In this regard, the research conducted by Mandelbrot and his colleagues reveals a new outlook in negative dimensions in geometry, see  , applications of which were found to turbulence, see Molenaar et al. 76 ,Chhabra and Sreenivasan 77,78 .Note that the negative dimensions described based on the Duplantier's function are from a view of geometry.Differing from the work by  this paper addresses time series or random functions with negative dimensions.It is well known that commonly used models of fractal time series are fractional Brownian motion fBm and fractional Gaussian noise fGn that is the increment process of fBm Mandelbrot 17 .For fBm as well as fGn, dimensions are restricted to be positive.As a matter of fact, denote by D fGn and H fGn the fractal dimension of fGn and its Hurst parameter, respectively.Denote fGn by x fGn t .Then, the autocorrelation function ACF of fGn follows The right side of the above expression requires 0 < H fGn < 1; see Mandelbrot 17 .Thus, 1 < D fGn < 2 because D fGn 2 − H fGn .In the LRD case, 0.5 < H fGn < 1 and 1.5 < D fGn < 2. As a result, I have the following note.
Note 1.The dimension of fGn is never negative.That is, 1 There are various types of fractal time series, such as fGn, alpha-stable processes, Levy processes.However, not all time series have negative dimensions.For example, fGn does not have negative dimensions.However, there may exist time series that have negative dimensions.This research of mine restricts my study to a specific class of Gaussian random functions.It is the extension of the GC process that were reported by Gneiting  Let D GC and H GC be the fractal dimension and the Hurst parameter of the GC process, respectively.Then, D GC and H GC are independent of each other.In the previous research regarding the GC process, only the case of 1 < D GC < 2 was discussed, see 79-81 , Li 52 , Li and Lim 83,84 .In this paper, I extend the GC process such that D EGC ∈ 0, 1 and D EGC < 0, where D EGC is the fractal dimension of the EGC process in addition to the traditional case of D EGC ∈ 1, 2 .The term EGC process means that it is a class of processes that is based on the GC process but extended to the case of D EGC < 1, simply for the purpose of distinguishing it from the standard GC process.
The rest of paper is organized as follows.The class of negatively dimensional random functions, that is, EGC processes, is addressed in Section 2. Discussions are given in Section 3. Extending the presented class of negative dimensional random functions to the corresponding random fields is briefed in Section 4. Finally, Section 5 concludes the paper.

Extended Generalized Cauchy (EGC) Processes
Denote by Ω, T, P the probability space.Then, x t, ζ is said to be a stochastic process when the random variable x represents the value of the outcome of an experiment T for every time t, where Ω represents the sample space, T is the event space or sigma algebra, and P the probability measure.
As usual, x t, ζ is simplified to be written as x t , that is, the event space is usually omitted.Denote by P x the probability function of x.Then, one can define the general nth order, time varying, joint distribution function P x 1 , .In this paper, only the first-and second-order properties of processes are considered instead of the complete joint distribution function.Moreover, this research only considers Gaussian processes.Gaussian processes can be completely determined by the second-order properties, more precisely, mean and ACF, see Papoulis 85 .Without generality losing, this paper only considers processes with mean zero.
Note that 1 |τ| α −β/α is a valid ACF for α > 0 and β > 0. Denote 1 |τ| α −β/α by r EGC τ that is the ACF of a Gaussian random function denoted by x EGC t .That is, Then, we call a random function x t as an EGC process if it is a stationary Gaussian centred process with the ACF given by 2.
where g ∈ S. Combining any r ∈ H EGC with its limit yields that H EGC is a Hilbert space.Denote it by For 0 < α ≤ 2, I express the subspace of H EGC by Denote by H GC the space of the standard GC process.Then, one immediately sees that H GC is a subspace of H EGC .More precisely, In the case of 2 < α ≤ 4, I denote another subspace of H EGC by Further, I denote the subspace of H EGC for α > 4 by Then, I have the remark below.Adler 96 , one says that a Gaussian stationary process is locally self-similar of order α if its ACF satisfies for τ → 0 Therefore, according to 93-96 , the fractal dimension denoted by D EGC2 is given by 0 This yields Theorem 2.4.
Let x EGC3 t be a process in H EGC3 .Denote the ACF of x EGC3 t by r EGC3 .Then, the following theorem gives the negative dimensions of x EGC3 t .Theorem 2.5.Processes in H EGC3 have negative dimensions.
Proof.Similar to 2.10 , one has Then, following 2.12 , we have This completes the proof of Theorem 2.5.
Remark 2.6.For an EGC process x EGC t with 0 < α and 0 < β, the fractal dimension of x EGC t in general satisfies

2.14
Theorem 2.7.The power spectrum density (PSD) function of the EGC process is given where ω is angular frequency, * implies the convolution, Sa ω sin ω /ω and

2.16
Proof.Note that 1 x ν can be expanded as a binomial series given by where ν ∈ Ê. and ν k is the binomial coefficient.Now, I expand the ACF of the EGC process by where u τ is the Heaviside unit step function Li and Lim 84 .
Because the Fourier transform FT of |t| λ is expressed by where λ / − 1, −3, . .., I have Similarly, Note that F u τ 1 − u τ − 1 2 Sa ω .Doing the FT of the first term on the right side of 2.18 term-by-term yields the following:

2.22
In addition, computing the FT of the second term on the right side of 2.18 term-byterm yields

2.23
Adding the right sides of 2.22 and 2.23 yields the result of this theorem.
Remark 2.8.The EGC processes are non-Markovian since r EGC t 1 , t 2 does not satisfy the triangular relation given by which is a necessary condition for a Gaussian process to be Markovian; see Todorovic 97 .In fact, up to a multiplicative constant, the Ornstein-Uhlenbeck process is the only stationary Gaussian Markov process; see Lim and Muniandy 98 , Wolperta and Taqqu 99 .
Note 1.Since r EGC τ ∼ |τ| −β for τ → ∞, one has the Hurst parameter of the EGC processes given by Note 2. Parameter α is independent of β and vice versa in the ACF of an EGC process.
Note 3. Dimensions of an EGC process relies on the value of α, irrelevant with β.That is, dimensions of an EGC process is irrelevant with its statistic dependenc.
I will discuss the meaning of D EGC2 and D EGC3 in the next section.

Discussions
The emphasized point I will explain is that the ACF, 2.2 , of the EGC process differs significantly from that of the GC process because I relax the restriction of α to be α > 0 instead of 0 < α ≤ 2 as that in the GC process, though two ACFs appear the similar, referring 80, 81 for the details of the GC process.Relaxing the range of α from 0 < α ≤ 2 in the GC model to α > 0 in this paper makes a considerable step further in the aspect of dimensions of random functions.To exhibit this step, I should explain the meaning of dimensions less than one and negative for a random function.
The fractal index α of a random function x t is considered for τ → 0 in 2.10 or in the following expression if the ACF of x t is sufficiently smooth on 0, ∞ : where α relates to D by D 1 − α/2.Obviously, r EGC is sufficiently smooth on 0, ∞ .As implied by 3.1 , one sees that the larger the value of α, the smoother the sample path of a random function.The following notes become apparent, accordingly.
Note 1.The present fractal dimensions, say D EGC2 and D EGC3 , imply that conventionally random functions are not the locally smoothest because there are random functions with dimensions less than one, for example, D EGC2 , or even negative, for example, D EGC3 .Note 2. The zero dimension occurs when α 4. That is, 3.2 Note 3. Random functions with D EGC 0 are not the locally smoothest.They are locally rougher than those with D EGC3 < 0.
In the extreme case of α → ∞, I have we say that the locally smoothest random function is that with D EGC → −∞.
Note 4. For D EGC → −∞, x EGC t is locally uncorrelated at any point of t.However, it may not be a white noise because it may globally be LRD if 0 < β < 1.As a matter of fact, there are the Hurst effects on x EGC t regardless of the value of D EGC .
Note 5.The standard GC process is a special case of the EGC process for 0 < α ≤ 2 and β > 0, which has applications to relaxation description in physics Lim and Li

3.10
In the discrete case, we have

3.11
where FFT represents the fast Fourier transform and IFFT stands for its inverse.Figure 1 indicates the realizations of the EGC process with various values of α for β 0.8 the LRD case .

Extension to Corresponding Random Field with Negative Dimension
Denote by Ê n the n-dimensional Euclidean space.The bold letters t and τ represent vectors belonging to Ê n , respectively denoting t t 1 , . . ., t n and τ τ 1 , . . ., τ n .The symbols t and τ represent their Euclidean norms.
In the work by Lim and Teo 105 , a random field X t is called a Gaussian field with the GC's covariance function if its covariance is given by R where β > 0 and α is restricted by 0 < α ≤ 2. Lim and Teo termed X t as the GFGCC Gaussian field with the generalized Cauchy covariance in short.We denote X t by X GFGCC t for simplicity.Over the hyperrectangle C n i 1 a i , b i , they obtained the positive fractal dimension of X GFGCC t by the following expression:   Clearly, n ≤ D GFGCC < n 1.

4.3
As previously discussed in Sections 2 and 3, the restriction of α can be relaxed to α > 0. Therefore, GFGCC can be extended to the case of α > 0. We call such an extension by EGFGCC extended Gaussian field with the generalized Cauchy covariance and denote it by X EGFGCC t .The fractal dimension of X EGFGCC t is expressed by The difference between X EGFGCC t and X GFGCC t is considerable because D EGFGCC may be negative while D GFGCC is always positive.As a matter of fact, we have The meaning of negative dimension for X EGFGCC t is similar to that explained in Section 3.That is, locally, X EGFGCC t is more regular for smaller D EGFGCC .

Conclusions
I have explained that the GC process can be extended to the EGC process with dimensions less than 1 and negative.The EGC process is rich such that it takes the standard GC process as its special case.I have explained that the EGC process with smaller fractal dimensions is smoother than that with larger ones.The present results are theoretic but data with negative dimensions have been synthesized in this paper.One interesting thing, as a consequence of this paper, is to explore such a class of data in various fields, for example, either in time series as those in 13, 14, 106-127 , or random fields 103 .
and Schlather 79 , Lim and Li 80 , Li and Lim 81 , and Li et al. 82 .The extended GC EGC processes may have negative dimensions.
EGC t in this paper.The norm r EGC τ is suitable for our research purpose.In fact, since x EGC t is Gaussian, it is uniquely determined by r EGC τ and vice versa.Note that Thus, I need to express r EGC τ in the domain of generalized functions.Definition 2.1 seeGriffel 91.A function of rapid decay is a smooth function φ : Ê → such that t n φ r t → 0 as t → ± ∞ for all n, r ≥ 0, where is the space of complex numbers.The set of all functions of rapid decay is denoted by S.
2 .Usually, the norm of a random function x t is expressed by E x t x t x t 2 , see, for example, Cramer 86, 87 , Liu 88 , Gelfand and Vilenkin 89 , and Adler et al. 90 .However, for the EGC process, E x EGC t x EGC t r EGC 0 1 regardless of the values of α and β.Therefore, it is inconvenient to use x EGC t .For this reason, I utilize r EGC τ rather Mathematical Problems in Engineering than x Lemma 2.2 see Griffel 91 .Every function belonging to S is absolutely integrable.Now, define the norm and inner product of r ∈ H EGC by EGC ∈ H EGC corresponds to a Gaussian process, see Gelfand and Vilenkin 89, Chapter 4 , Kanwal 92 .Its ACF is given by 2.2 .However, the dimensions of processes in H EGC2 and H EGC3 are undefined, more precisely, unknown.The following theorems will describe the dimensions of processes in H EGC2 and H EGC3 .Processes in H EGC2 have the dimensions less than one and greater than or equal to zero.Proof.Denote a process in H EGC2 by x EGC2 t .Let an ACF in H EGC2 be r EGC2 .Then, taking into account the definition of the local self-similarity provided by Kent and Wood 93 , Hall and Roy 94 , Chan et al. 95 , Remark 2.3.One hasH EGC H EGC1 ∪ H EGC2 ∪ H EGC3 .Recall that each r 80 , the Internet traffic Li and Lim 81 , Li and Zhao 100 , chromatin morphologies in breast cancer cells Muniandy and Stanslas 101 .Note 6.The usual Cauchy process is a special case of the EGC process.In fact, when α β 2, one gets the ACF of the usual Cauchy process.Denote by r C τ the ACF of the usual Cauchy process.Then, It is easily seen that the fractal dimension of the usual Cauchy process is one.It is SRD since r C τ ∼ |τ| −2 for τ → ∞.In what follows, I denote r EGC τ by r EGC τ; α, β for facilitating the explanation.When α 2 and β 1, one has The above r EGC τ; 2, 1 is the ACF proposed by Spector and Grant 102 for interpreting areomagnetic data, which is a special case of the EGC process.When α 2 and β 3, r EGC τ At moment, I am unaware what practical data may have the fractal dimensions less than one or negative but we are able to synthesize such data following the simulation method by Li 104 .Denote by w t the standard white noise.Let F −1 be the operator of the inverse Fourier transform.Denote by y t the synthesized random function that follows the ACF of the EGC process.Then, Replacing α and β by D EGC and H EGC , respectively, yields 8, that is, H EGC 0.6.a Realization for α 0.1, that is, D EGC 1.95.b Realization for α 1, that is, D EGC 1.5.c Realization for α 2, that is, D EGC 1. d Realization for α 4, that is, D EGC 0. e Realization for α 8, that is, D EGC −2.f Realization for α 16, that is, D EGC −6.