Generalization of the Khinchin Theorem to Levy Flights

One of the most fundamental theorems in statistical mechanics is the Khinchin ergodic theorem, which links the ergodicity of a physical system with the irreversibility of the corresponding autocorrelation function. However, the Khinchin theorem cannot be successfully applied to processes with infinite second moment, in particular, to the relevant class of Levy flights. Here, we solve this challenging problem. Namely, using the recently developed measure of dependence called Levy correlation cascade, we derive a version of the Khinchin theorem for Levy flights. This result allows us to verify the Boltzmann hypothesis for systems displaying Levy-flight-type dynamics.