EVOLUTION TOWARDS ERGODIC BEHAVIOR OF STATIONARY FRACTAL RANDOM-PROCESSES WITH MEMORY - APPLICATION TO THE STUDY OF LONG-RANGE CORRELATIONSOF NUCLEOTIDE-SEQUENCES IN DNA

The possible occurrence of ergodic behavior for large times is investi gated in the case of stationary random processes with memory. It is sh own that for finite times the time average of a state function is gene rally a random variable and thus two types of cumulants can be introdu ced: for the time average and for the statistical ensemble, respective ly. 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 cumulan ts is developed; explicit computations are presented both for short an d long memory in the particular case of separable stationary processes for which the cumulants of a statistical ensemble can be factorized i nto products of functions depending on binary time differences. In bot h cases the ergodic behavior emerges for large times provided that the cumulants of a statistical ensemble decrease to zero as the time diff erences increase to infinity. The analysis leads to the surprising con clusion 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 p ower scaling laws. If the cumulants of a statistical ensemble tend tow ards asymptotic values different from zero for large time differences, then the time average is random even as the length of the total time interval tends to infinity and the ergodic behavior no longer holds. T he theory is applied to the study of long range correlations of nucleo tide sequences in DNA; in this case the length t of a sequence of nucl eotides plays the role of the time variable. A proportionality relatio nship 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 th e statistical analysis of the DNA data presented in the literature is consistent with the occurrence of the ergodic behavior for large lengt hs. The implications of the approach to the analysis of the large time behavior of stochastic cellular automata and of fractional Brownian m otion are also investigated.