title: Lévy random walks with fluctuating step number and multiscale behavior.
abstract: Random walks with step number fluctuations are examined in n dimensions for when step lengths comprising the walk are governed by stable distributions, or by random variables having power-law tails. When the number of steps taken in the walk is large and uncorrelated, the conditions of the Lévy-Gnedenko generalization of the central limit theorem obtain. When the number of steps is correlated, infinitely divisible limiting distributions result that can have Lévy-like behavior in their tails but can exhibit a different power law at small scales. For the special case of individual steps in the walk being Gaussian distributed, the infinitely divisible class of K distributions result. The convergence to limiting distributions is investigated and shown to be ultraslow. Random walks formed from a finite number of steps modify the behavior and naturally produce an inner scale. The single class of distributions derived have as special cases, K distributions, stable distributions, distributions with power-law tails, and those characteristic of high and low frequency cascades. The results are compared with cellular automata simulations that are claimed to be paradigmatic of self-organized critical systems.
authors.affiliation: Theoretical Mechanics Division, School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom.
authors.forename: K I
authors.initials: KI
authors.lastname: Hopcraft
authors.forename: E
authors.initials: E
authors.lastname: Jakeman
authors.forename: R M
authors.initials: RM
authors.lastname: Tanner