A new stochastic renormalization approach to random processes with very long memory: fractal time as a process with 'almost' complete connections

An attempt is made to apply the mathematical theory of random jump processes with complete connections to the physically oriented description of very long memory effects. A new stochastic renormalization approach is introduced. Starting from a stochastic process with infinite memory, the author attaches to each step a very small probability that the memory is lost. As a result of this change the memory acts over a variable number of steps, forming 'blocks' of different sizes. This renormalization procedure leads to the self-similarity of the probability of the number of steps over which the memory is kept. The model yields a new explanation for the fractal time.