All of the above functions are true fractals in that scaling is exhibited for increment distributions over all lags from zero to infinity. In the present study, we examine the development of new fractals and empirical scaling models based on PDF'ss that are not members of the Levy-stable family. Most applications of the Levy model require truncation of the tails of the distribution which ultimately become too fat because of the power law die off.
Truncation also causes the distribution that is actually used to have a finite variance.
Gaussian distributions have tails that decrease too rapidly. While an empirical scaling model may be developed based on the Laplace PDF, the distribution is not closed under convolution and so does not produce theoretically a fractal with consistent scaling over an infinite lag range. Nevertheless, we argue that such a model may still be useful since the property distributions themselves don't appear to be true fractals. What seems ubiquitous in nature is scaling over a finite range, and this appears to be reproducible with what we are calling a Laplace scaling model. The Laplace PDF is a limiting member of the Gamma family of PDF's, and this family is closed under convolutions and so can serve as the basis for the development of a set of infinite scaling models.