Rates of Convergence in Ergodic Theorems
by
Alexander Kachurovskii
Steklov Institute of Mathematics at St.Petersburg

Let A_nf be usial n-th ergodic average of summable function f; and let these A_nf converge to their limit function f^. Then this convergence (a.e.) is equivalent to such a property:
The sequence P(n, varepsilon)=P(A_Nf-f^|>= varepsilon for some N>=n") tends to 0 as n tends to infty, for every varepsilon >0.
As it occures, in the case f in L_2, this tend can be estimated via the singularity of the spectral measure in zero point; or via the speed of decay of correlation coefficients; or via the speed of decay of variances of A_nf.
The main results here, and historical comments, are published in Uspehi Mat. Nauk, 1996, vol.51, p.73-124 (in Russian; English translation in: Russian Math. Surveys, 1996, vol.51, p.653-703).
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There is a strange similarity in the behaviour of the ergodic averages and (reversed) martingales; it is well-known since early 1950-th (Kakutani, Doob). In the second part of the talk, it will be suggested one more approach of explanation of this interesting effect. As it occures, both ergodic theorems and martingales can be considered as partial cases of the one general class of stochastic processes.
Let F_n be a monotone (monotone increasing or monotone decreasing) sequence of sigma-subfields of the main sigma-field of our probability space (X, F, P); and let these F_n tends to F_infty. Then:
Conditional expectations E(A_nf|F_n) converge (a.e. and in the norm) to E(f^|F_infty).
In the case all F_n coinside with F - this theorem is usial ergodic theorem (with some obvious restrictions). In the case automorphism T is identity - all A_nf coinside with f, and the theorem above is usial martingale convergence theorem (with some restrictions, too).
Proofs and further comments are published in Mat. Zametki, 1998, vol.64, p.311-314 (in Russian; English translation: to appear in Math. Notes, 1998, vol.64).

Date received: July 5, 1999