Evolution towards ergodic behavior of stationary fractal random processes with memory: application to the study of long-range correlations of nucleotide sequences in DNA

The possible occurrence of ergodic behavior for large times is investigated in the case of stationary random processes with memory. It is shown that for finite times the time average of a state function is generally a random variable and thus two types of cumulants can be introduced: for the time average and for the statistical ensemble, respectively. In the limit of infinite time a transition from the random to the deterministic behavior of the time average may occur, resulting in an ergodic behavior. The conditions of occurrence of this transition are investigated by analyzing the scaling behavior of the cumulants of the time average. A general approach for the computation of these cumulants is developed; explicit computations are presented both for short and long memory in the particular case of separable stationary processes for which the cumulants of a statistical ensemble can be factorized into products of functions depending on binary time differences. In both cases the ergodic behavior emerges for large times provided that the cumulants of a statistical ensemble decrease to zero as the time differences increase to infinity. The analysis leads to the surprising conclusion that the scaling behavior of the cumulants of the time average is relatively insensitive to the type of memory considered: both for short and long memory the cumulants of the time average obey inverse different from zero for large time differences, then the time averaage is random even as the length of the total time interval tends to infinity and the ergodic behavior no longer holds. The theory is applied to the study of long range correlations of nucleotide sequences in DNA; in this case the length t of a sequence of nucleotides plays the role of the time variable. A proportionality relationship is established between the cumulants of the pyrimidine excess in a sequence of length t and the cumulants of the time (length) average of the probability of occurrence of a pyrimidine. It is shown that the statistical analysis of the DNA data presented in the literature is consistent with the occurrence of the ergodic behavior for large lengths. The implications of the approach to the analysis of the large time behavior of stochastic cellular automata and of fractional Brownian motion are also investigated.