Paired Levy-Mandelbrot trajectory as a homogeneous fractal

This paper deals with the problem of using the Levy-Mandelbrot (LM) trajectory for constructing a homogeneous random point fractal in three-dimensional space. For investigating stochastic properties of a random number of points M(R) within a radius R around an arbitrary occupied point, it is proposed to use a paired LM trajectory. It is found that for sufficiently large values of R the distribution of the random variable M(R)/[M(R)] does not depend on R and can be approximated by the gamma distribution with a unique parameter depending on the fractal dimension. This property is analogous to KNO scaling of multiplicity in hadron-hadron collisions.