Random Multiplicative Processes and Multifractals

Abstract

Random multiplicative processes abound in a variety of natural phenomena./l,2/ Specific examples include the distribution of incomes, body weights, rainfall, etc., in addition to the examples that will be treated in this paper. While random multiplicative processes are relatively ubiquitous, the essential properties of such processes are not as well appreciated as random additive processes (such as a random walk). For the latter case, the existence of the central limit theorem provides crucial information for understanding the asymptotic properties of a sum of random variables. From this theorem, one knows that short-range correlations in the sequence of random variables do not affect the asymptotic properties of the sum, and that one-parameter scaling holds. On the other hand, multiplicativity gives rise to multifractal scaling, and this property has been seen in a wide variety of contexts./3-9/