Randomly amplified discrete Langevin systems.

A discrete stochastic process involving random amplification with additive noise is studied analytically. If the non-negative random amplification factor b is such that =1, where beta is any positive noninteger, then the steady state probability density function for the process will have power law tails of the form p(x) approximately 1/x(beta+1). This is a generalization of recent results for 0<beta<2 obtained by Takayasu, Sato, and Takayasu [Phys. Rev. Lett. 79, 966 (1997)]. It is shown that the power spectrum of the time series x becomes Lorentzian, even when 1<beta<2, i.e., in the case of divergent variance.