Geodesic
General Theories Unifying Ergodic Averages and Martingales
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and optimization, Tome 256 (2007), pp. 172-200

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The curious fact that the behavior of ergodic averages is analogous to the behavior of (reversed) martingales has long been known. For the last 60 years, at least five different approaches have been proposed to the construction of a unifying theory: by M. Jerison (1955), G. C. Rota (1961), A. and C. Ionescu Tulcea (1963), A.M. Vershik (1960s), and A.G. Kachurovskii (1998). The aim of the present survey is to carry out a comparative analysis of all the approaches, with special focus being placed on the fifth approach, which is due to the present author and has not yet been published in full detail. Moreover, the comparative analysis carried out in the survey has led to a new, the sixth, approach. 
@article{TM_2007_256_a8,
     author = {A. G. Kachurovskii},
     title = {General {Theories} {Unifying} {Ergodic} {Averages} and {Martingales}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {172--200},
     publisher = {mathdoc},
     volume = {256},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2007_256_a8/}
}
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A. G. Kachurovskii. General Theories Unifying Ergodic Averages and Martingales. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and optimization, Tome 256 (2007), pp. 172-200. http://geodesic.mathdoc.fr/item/TM_2007_256_a8/