Growing random sequences
Ilia Krasikov, Geoff J. Rodgers, C E Tripp · Journal of Physics A Mathematical and General · 2004
We consider the random sequence x n = x n −1 + γ x q , with γ > 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 2 n ⟩ as functions of n and γ. For γ = 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) .