Growing random sequences

Abstract

We consider the random sequence x[n] = x[n-1] + yxq, with y > 0, where q = 0, 1,..., n - 1 is chosen randomly from a probability distribution P[n] (q). When all q are chosen with equal probability, i.e. P[n](q) = 1/n, we obtain an exact solution for the mean <x[n]> and the divergence of the second moment <x[n]2> as functions of n and y. For y = 1 we examine the divergence of the mean value of x[n], as a function of n, for the random sequences generated by power law and exponential P[n](q) and for the non-random sequence P[n](q) = δ[q,β(n-1)].