A Class of Negatively Fractal Dimensional Gaussian Random Functions

Abstract

Letx(t)be a locally self-similar Gaussian random function. Denote byrxx(τ)the autocorrelation function (ACF) ofx(t). Forx(t)that is sufficiently smooth on(0,∞), there is an asymptotic expression given byrxx(0)-rxx(τ)~c|τ|αfor|τ|→0, wherecis a constant andαis 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,αis strictly restricted to0<α≤2so 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 indexαofx(t)may be relaxed to the rangeα>0as long as its ACF keeps valid forα>0. With this claim, I extend the GC process to allowα>0and call this extension, for simplicity, the extended GC (EGC for short) process. I will address that there are dimensions0≤D(α)<1for2<α≤4and furtherD(α)<0for4<αfor the EGC processes. I will explain thatx(t)with1≤D<2is locally rougher than that with0≤D<1. Moreover,x(t)withD<0is 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<1remain 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.