Aaronson, Jon; Denker, Manfred; Fisher, Albert M. Second order ergodic theorems for ergodic transformations of infinite measure spaces. (English) Zbl 0738.28011 Proc. Am. Math. Soc. 114, No. 1, 115-127 (1992). Summary: For certain pointwise dual ergodic transformations \(T\) we prove almost sure convergence of the log-averages \[ {1 \over \log N}\sum_{n=1}^ N {1 \over na(n)} \sum_{k=1}^ n f\circ T^ k \qquad (f\in L_ 1) \] and the Chung-Erdős averages \[ {1 \over \log a(N)} \sum_{k=1}^ N {1 \over a(k)} f\circ T^ k \qquad (f\in L_ 1^ +) \] towards \(\int f\), where \(a(n)\) denotes the return sequence of \(T\). Cited in 1 ReviewCited in 7 Documents MSC: 28D05 Measure-preserving transformations 60F15 Strong limit theorems Keywords:measure-preserving dynamical system; Markov shift; second order ergodic theorems; infinite measure spaces; pointwise dual ergodic transformations; almost sure convergence; log-averages; Chung-Erdős averages; return sequence PDFBibTeX XMLCite \textit{J. Aaronson} et al., Proc. Am. Math. Soc. 114, No. 1, 115--127 (1992; Zbl 0738.28011) Full Text: DOI References: [1] Jon Aaronson, Ergodic theory for inner functions of the upper half plane, Ann. Inst. H. Poincaré Sect. B (N.S.) 14 (1978), no. 3, 233 – 253 (English, with French summary). · Zbl 0378.28009 [2] Jon Aaronson, The asymptotic distributional behaviour of transformations preserving infinite measures, J. Analyse Math. 39 (1981), 203 – 234. · Zbl 0499.28013 · doi:10.1007/BF02803336 [3] Jon. Aaronson, Random \?-expansions, Ann. Probab. 14 (1986), no. 3, 1037 – 1057. · Zbl 0658.60050 [4] T. Bedford and A. M. Fisher, Analogues of the Lebesgue density theorem for fractal subsets of the reals and integers, Proc. London Math. Soc. (to appear). · Zbl 0706.28009 [5] K. L. Chung and P. Erdös, Probability limit theorems assuming only the first moment. I, Mem. Amer. Math. Soc., No. 6 (1951), 19. · Zbl 0042.37601 [6] William Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. · Zbl 0077.12201 [7] A. M. Fisher, A pathwise central limit theorem for random walks, Ann. Prob. (to appear). [8] -, Integer Cantor sets and an order-two ergodic theorem, J. D’Analyse Math. (to appear). · Zbl 0788.58032 [9] Gérard Letac, Which functions preserve Cauchy laws?, Proc. Amer. Math. Soc. 67 (1977), no. 2, 277 – 286. · Zbl 0376.28019 [10] Maximilian Thaler, Transformations on [0,1] with infinite invariant measures, Israel J. Math. 46 (1983), no. 1-2, 67 – 96. · Zbl 0528.28011 · doi:10.1007/BF02760623 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.