Randomly amplified discrete Langevin systems

A discrete stochastic process involving random amplification with additive noise is studied analytically. If the non-negative random amplification fac tor b is such that [b(beta)] = 1, where beta is any positive noninteger, th en the steady state probability density function for the process will have power law tails of the form p(x) similar to 1/x(beta+1). This is a generali zation of recent results for 0<beta<2 obtained by Takayasu, Sate, and Takay asu [Phys. Rev. Lett. 79, 966 (1997)]. Iris shown that the power spectrum o f the time series x becomes Lorentzian, even when 1<beta<2, i.e., in the ca se of divergent variance. [S1063-651X(99)0306-4].