A Class of Negatively Fractal Dimensional Gaussian Random Functions

Let x ( t ) be a locally self-similar Gaussian random function. Denote by r x x ( τ ) the autocorrelation function (ACF) of x ( t ) . For x ( t ) that is sufficiently smooth on ( 0 , ∞ ) , there is an asymptotic expression given by r x x ( 0 ) - r x x ( τ ) ~ c | τ | α 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 of D ∈ [ 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 ) with D < 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 for D < 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.