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 $<b^{\beta}>=1$ where $\beta$ is any positive non-integer, then the steady state probability density function for the process will have power law tails of the form $p(x) \sim 1/x^{\beta +1}$. This is a generalization of recent results for $0 < \beta < 2$ obtained by Takayasu et al. in 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 case of divergent variance.