A Class of Negatively Fractal Dimensional Gaussian Random Functions. (30th November 2010) - Li, Ming
Abstract : Letx ( t ) be a locally self-similar Gaussian random function. Denote byr x x ( τ ) the autocorrelation function (ACF) ofx ( t ) . Forx ( t ) that is sufficiently smooth on( 0, ∞ ), there is an asymptotic expression given byr x x ( 0 ) - r x x ( τ ) c | τ | α for| τ | → 0, wherec is a constant andα is the fractal index ofx ( t ) . If the above is true, the fractal dimension ofx ( t ), denoted byD, is given byD = D ( α ) = 2 − α / 2 . Conventionally, α is strictly restricted to0 < α ≤ 2 so as to make sure thatD ∈ [ 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 ) or0 < α ≤ 2 . In this paper, I claim that the fractal indexα ofx ( t ) may be relaxed to the rangeα > 0 as long as its ACF keeps valid forα > 0 . With this claim, I extend the GC process to allowα > 0 and call this extension, for simplicity, the extended GC (EGC for short) process. I will address that there are dimensions0 ≤ D ( α ) < 1 for2 < α ≤ 4 and furtherD ( α ) < 0 for4 < α for the EGC processes. I will explain thatx ( t ) with1 ≤ D < 2 is locally rougher than that with0 ≤ D < 1 . Moreover, x ( t ) withD < 0 is locally smoother than that with0 ≤ D < 1 . The local smoothestx ( t ) occurs in the limitD → − ∞ . 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 forD < 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.