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A Class of Negatively Fractal Dimensional Gaussian Random Functions

机译：一类负分形维数高斯随机函数

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Let x(t) be a locally self-similar Gaussian random function. Denote by r_(xx)(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_(xx)(0) - r_(xx)(τ) ~ c|τ|~a for |τ| →0, where c is a constant and a 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 - a/2. Conventionally, a 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 a 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.
机译：令x（t）是局部自相似的高斯随机函数。用r_（xx）（τ）表示x（t）的自相关函数（ACF）。对于在（0，∞）上足够平滑的x（t），存在一个关于|τ|的由r_（xx）（0）-r_（xx）（τ）〜c |τ|〜a给出的渐近表达式。 →0，其中c是常数，a是x（t）的分形指数。如果上述条件成立，则x（t）的分形维数用D表示：D = D（α）= 2-a / 2。按照惯例，为了确保D∈[1,2]，a被严格限制为0 <α<2。广义柯西（GC）过程是此类随机函数的一个实例。另一个例子是分数布朗运动（fBm）及其增量过程，即分数高斯噪声（fGn），严格遵循D∈[1,2]或0 <α≤2的情况。在本文中，我主张只要它的ACF对α> 0保持有效，就可以将x（t）的分形指数a放宽到α> 0的范围。基于这一主张，我扩展了GC的过程，使α> 0，并称其为扩展，为简单起见，扩展了GC（简称ECC）过程。我要解决的是，对于EGC过程，尺寸2 <α<4的尺寸为D（α）<1，对于4 <α的尺寸为D（α）<0。我将解释1

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    《Mathematical Problems in Engineering》 |2011年第2期|p.1-18|共18页
  • 作者

    Ming Li;

  • 作者单位

    School of Information Science & Technology, East China Nonnal University, No. 500, Dong-Chuan Road, Shanghai 200241, China;

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