Levy random walks with fluctuating step number and multiscale behavior

Random walks with step number fluctuations are examined in it dimensions fo r when step lengths comprising the walk are governed by stable distribution s, 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 Levy-Gne denko generalization of the central limit theorem obtain. When the number o f steps is correlated, infinitely divisible limiting distributions result t hat can have Levy-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 distri butions result. The convergence to limiting distributions is investigated a nd 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 di stributions, distributions with power-law tails, and those characteristic o f high and low frequency cascades. The results are compared with cellular a utomata simulations that are claimed to be paradigmatic of self-organized c ritical systems. [S1063-651X(99)11511-3].