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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - A. G. Kachurovskii
TI  - General Theories Unifying Ergodic Averages and Martingales
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2007
SP  - 172
EP  - 200
VL  - 256
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2007_256_a8/
LA  - ru
ID  - TM_2007_256_a8
ER  - 
%0 Journal Article
%A A. G. Kachurovskii
%T General Theories Unifying Ergodic Averages and Martingales
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2007
%P 172-200
%V 256
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2007_256_a8/
%G ru
%F TM_2007_256_a8
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/

[1] Vershik A.M., “Teoriya ubyvayuschikh posledovatelnostei izmerimykh razbienii”, Algebra i analiz, 6:4 (1994), 1–68 | MR | Zbl

[2] Vershik A.M., Kachurovskii A.G., Skorosti skhodimosti v ergodicheskikh teoremakh dlya lokalno konechnykh grupp i obraschennye martingaly, Preprint POMI RAN 2/1998, 6 pp. ; Диф. уравнения и процессы управления, No 1, 1999, 19–26 | Zbl | MR

[3] Danford N., Shvarts Dzh., Lineinye operatory: Obschaya teoriya, Izd-vo inostr. lit., M., 1962

[4] Dub Dzh.L., Veroyatnostnye protsessy, Izd-vo inostr. lit., M., 1956 | MR

[5] Ibragimov I.A., Linnik Yu.V., Nezavisimye i statsionarno svyazannye velichiny, Nauka, M., 1965

[6] Ivanov V.V., “Geometricheskie svoistva monotonnykh funktsii i veroyatnosti sluchainykh kolebanii”, Sib. mat. zhurn., 37:1 (1996), 117–150 | MR | Zbl

[7] Kachurovskii A.G., “Skorosti skhodimosti v ergodicheskikh teoremakh”, UMN, 51:4 (1996), 73–124 | MR

[8] Kachurovskii A.G., “Martingalno-ergodicheskaya teorema”, Mat. zametki, 64:2 (1998), 311–314 | MR | Zbl

[9] Kachurovskii A.G., “O skhodimosti srednikh v ergodicheskoi teoreme dlya grupp $\mathbb Z^d$”, Zap. nauch. sem. POMI, 256 (1999), 121–128 | MR

[10] Kornfeld I.P., Sinai Ya.G., Fomin S.V., Ergodicheskaya teoriya, Nauka, M., 1980 | MR

[11] Prigozhin I., Stengers I., Poryadok iz khaosa, Progress, M., 1986

[12] Rokhlin V.A., “Ob osnovnykh ponyatiyakh teorii mery”, Mat. sb., 25:1 (1949), 107–150 | Zbl

[13] Stepin A.M., “Entropiinyi invariant ubyvayuschikh posledovatelnostei izmerimykh razbienii”, Funkts. analiz i ego pril., 5:3 (1971), 80–84 | MR | Zbl

[14] Khalmosh P., Lektsii po ergodicheskoi teorii, Izd-vo inostr. lit., M., 1959

[15] Khardi G., Raskhodyaschiesya ryady, Izd-vo inostr. lit., M., 1951

[16] Khyuitt E., Ross K., Abstraktnyi garmonicheskii analiz, t. 1, 2, Mir, M., 1975

[17] Shiryaev A.N., Veroyatnost, Nauka, M., 1989 | MR

[18] Argiris G., Rosenblatt J.M., “Forcing divergence when the supremum is not integrable”, Positivity, 10:2 (2006), 261–284 | DOI | MR | Zbl

[19] Bellow A., “For the historical record”, Measure theory (Oberwolfach, 1983), Lect. Notes Math., 1089, Springer, Berlin, 1984, 271 | MR

[20] Bishop E., “An upcrossing inequality with applications”, Michigan Math. J., 13:1 (1966), 1–13 | DOI | MR | Zbl

[21] Blackwell D., Dubins L.E., “A converse to the dominated convergence theorem”, Ill. J. Math., 7:3 (1963), 508–514 | MR | Zbl

[22] Blake L.H., “Martingales, the smoothing index and ergodic theory”, Stoch. Processes and Appl., 4:2 (1976), 149–155 | DOI | MR | Zbl

[23] Blum J., Hanson D., “On the mean ergodic theorem for subsequences”, Bull. Amer. Math. Soc., 66:4 (1960), 308–311 | DOI | MR | Zbl

[24] Bourgain J., “Pointwise ergodic theorems for arithmetic sets”, Publ. Math. IHES, 69 (1989), 5–45 | MR | Zbl

[25] Bray G., “Théoremes ergodiques de convergence en moyenne”, J. Math. Anal. and Appl., 25:3 (1969), 471–502 | DOI | MR | Zbl

[26] Burkholder D.L., “Distribution function inequalities for martingales”, Ann. Probab., 1:1 (1973), 19–42 | DOI | MR | Zbl

[27] Burkholder D.L., “Successive conditional expectations of an integrable function”, Ann. Math. Stat., 33:3 (1962), 887–893 | DOI | MR | Zbl

[28] Derriennic Y., Lin M., “On invariant measures and ergodic theorems for positive operators”, J. Funct. Anal., 13:3 (1973), 252–267 | DOI | MR | Zbl

[29] Dubins L.E., “Rises and upcrossings of nonnegative martingales”, Ill. J. Math., 6:2 (1962), 226–241 | MR | Zbl

[30] Goubran N., “Pointwise inequalities for ergodic averages and reversed martingales”, J. Math. Anal. and Appl., 290:1 (2004), 10–20 | DOI | MR | Zbl

[31] Ionescu Tulcea A., Ionescu Tulcea C., “Abstract ergodic theorems”, Trans. Amer. Math. Soc., 107:1 (1963), 107–124 | DOI | MR

[32] Jerison M., “Martingale formulation of ergodic theorems”, Proc. Amer. Math. Soc., 10:4 (1959), 531–539 | DOI | MR | Zbl

[33] Jones R.L., Kaufman R., Rosenblatt J.M., Wierdl M., “Oscillation in ergodic theory”, Ergod. Theory and Dyn. Syst., 18:4 (1998), 889–935 | DOI | MR | Zbl

[34] Jones R.L., Rosenblatt J.M., Wierdl M., “Counting in ergodic theory”, Canad. J. Math., 51:5 (1999), 996–1019 | MR | Zbl

[35] Kakutani S., “Ergodic theory”, Proc. Intern. Congr. Math. (Cambridge (MA), 1950), 2, Amer. Math. Soc., Providence (RI), 1952, 128–142 | MR

[36] Krengel U., Ergodic theorems, W. de Gruyter, Berlin–New York, 1985 | MR | Zbl

[37] Maker P.T., “The ergodic theorem for a sequence of functions”, Duke Math. J., 6:1 (1940), 27–30 | DOI | MR | Zbl

[38] Neveu J., “Rélations entre la théorie des martingales et la théorie ergodique”, Ann. Inst. Fourier, 15:1 (1965), 31–42 | MR | Zbl

[39] Neveu J., Discrete-parameter martingales, North-Holland, Amsterdam, 1975 | MR | Zbl

[40] Ornstein D., “A remark on the Birkhoff ergodic theorem”, Ill. J. Math., 15:1 (1971), 77–79 | MR | Zbl

[41] Rao M.M., “Abstract martingales and ergodic theory”, Multivariational analysis. III, Proc. Third Intern. Symp. (Wright State Univ., Dayton (Ohio), 1972), Acad. Press, New York, 1973, 45–60 | MR

[42] Rao M.M., Foundations of stochastic analysis, Acad. Press, New York, 1981 | MR

[43] Rota G.-C., “Une théorie unifiee des martingales et des moyennes ergodiques”, C. r. Acad. sci. Paris, 252:14 (1961), 2064–2066 | MR | Zbl

[44] Rota G.-C., “Reynolds operators”, Proc. Symp. Appl. Math., 16 (1964), 70–83 | MR | Zbl

[45] Sucheston L., “On one-parameter proofs of almost sure convergence of multiparameter processes”, Ztschr. Wahrscheinlichkeitsth. und verw. Geb., 63:1 (1983), 43–49 | DOI | MR | Zbl

[46] Vershik A.M., “Amenability and approximation of infinite groups”, Sel. Math. Sov., 2:4 (1982), 311–330 | MR | Zbl

[47] Wiener N., “The ergodic theorem”, Duke Math. J., 5:1 (1939), 1–18 | DOI | MR | Zbl