Random process, ergodic

Cross-Correlation Function If n (t) and ng t) are two ergodic random processes it is possible to define the cross-correlation function by... 

Thus, we have seen that every binary DMS possesses the AEP in fact, so does every DMS. An information source need not be memoryless, though, to possess the AEP every stationary ergodic source—that is, a discrete source for which the output is a stationary ergodic random process—possesses the AEP (and, therefore, an entropy too). Even some nonstationary sources possess the AEP. For example, if the source sequence (wi, M2,. ..) is a time-invariant Markov chain, then its entropy is given by the formula... 

There have been a few interesting fundamental papers on ergodicity 42 45 The distinction was emphasized between true equilibrium and quasiequilibrium phenomena, which now are frequently observed in NMR. An isolated finite system should not be expected to become ergodic. Two other fundamentally interesting processes that have been demonstrated with solid-state NMR are dephasing caused by randomization of geometric phase,46 47 and the possibility of the chaotic behaviour of spin systems.48,49... 

Under certain circumstances it can be shown that in the limit of large number of iterations of the process, the random variable X tends to X (in distribution). Thus the Monte-Carlo Markov Chain method can be used to generate sets of independent realizations of the random variable which can be used to calculate an expectation. In practice, one prefers to use a long sequence of the iterates from a single starting value. The efficacy (specifically the ergodicity and rate of convergence) of the Monte-Carlo Markov-Chain method depends on the choice of the transition density. 

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