ABSTRACT
The study of spatial distributions that arise from independent multiplicative cascades can be traced back at least to the efforts of the Russian school led by Andrei N. Kolmogorov on the statistical theory of turbulence. However, largely due to the intriguing statistical/geometric scaling properties so heavily emphasized by Benoit Mandelbrot and to the rich mathematical foundations begun by Jean-Pierre Kahane and Jacques Peyrière in the middle 1970s, there has been a growing interest in random cascades both in the physical sciences and in mathematics. After a brief summary of some of the principal mathematical aspects of the theory of independent cascades, this paper provides some refinements and extensions of the theory in new directions. In particular, we provide a proof of a general composition theorem that provides the basis for a percolation method originally formulated by Kahane to compute carrying dimensions. Size biasing and the law of the iterated logarithm are then used to obtain estimates on the exact Hausdorff dimension of the support. Finally, a new class of correlated cascades that naturally possess a rich “multifractal structure” in the sense of the dimension disintegration theory of Cutler and Kahane and Katznelson is introduced.
