Random multiplicative processes: An elementary tutorial

An elementary discussion of the statistical properties of the product of N independent random variables is given. The motivation is to emphasize the essential differences between the asymptotic N→∞ behavior of a random product and the asymptotic behavior of a sum of random variables—a random additive process. For this latter process, it is widely appreciated that the asymptotic behavior of the sum and its distribution is provided by the central limit theorem. However, no such universal principle exists for a random multiplicative process. In this case, the ratio between the average value of the product 〈P〉 and the most probable value Pmp diverges exponentially in N as N→∞. Within a continuum approximation, the classical log-normal form is often invoked to describe the distribution of the product. It is shown, however, that the log-normal provides a poor approximation for the asymptotic behavior of the average value and, also, for the higher moments of the product. A procedure for computing the correct leading asymptotic behavior of the moments is outlined. The implications of these results for simulations of random multiplicative processes are also discussed. For such a simulation, the numerically observed ‘‘average’’ value of the product is of the order of Pmp, and it is only when the simulation is large enough to sample a finite fraction of all the states in the system that a monotonic crossover to the true average value 〈P〉 occurs. An idealized, but quantitative account for this crossover is provided.