Abstract
We give a new direct proof of the a.s. convergence of the Cesàro-α means of a stationary process (X n) when 0<α<1 andE(⋎X n⋎p)<+∞ with αp>1 and we show that this result does not hold in general for αp=1. We also consider similar questions for orthogonal random variables. Finally, we study the a.s. convergence of Riesz harmonic means.
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References
Akcoglu, M. A., and Del Junco, A. (1975). Convergence of averages of point transformations.Proc. Am. Math. Soc. 49, 265–266.
Chacon, R. (1964). A class of linear transformation.Proc. Am. Math. Soc. 15, 560–564.
Chow, Y. S., and Lai T. L. (1973). Limiting behaviour of weighted sums of independent random variables.Ann. Prob. 1(5), 810–824.
Chung, K. L. (1974).A Course in Probability, 2nd edition. Academic Press, New York.
Deniel, Y., and Derriennic, Y. (1988). Sur la convergence presque sûre, au sens de Cesàro d'ordre α, 0<α<1, de variables aléatoires indépendantes et identiquement distribuées.Prob. Theor. Rel. Fields 79, 629–636.
Hardy, G. H. (1949).Divergent Series, Oxford Univ. Press, London.
Irmisch, R. (1980). Punktweise Ergodensätze für (c,α)-Verfahren, 0<α<1. Thèse de doctorat, Darmstadt.
Krengel, U. (1985).Ergodic Theorems. De Gruyter, Berlin.
Lai, T. L. (1974). Summability methods for i.i.d. random variables.Proc. Am. Math. Soc. 43(2), 253–261.
Lorentz, G. G. (1955). Borel and Banach properties of methods of summation.Duke Math. J. 22, 129–141.
Wiener, N. (1939). The ergodic theorem.Duke Math. J. 5, 1–18.
Zygmund, A. (1968).Trigonometric Series, 2nd edition. Cambridge University Press, New York.
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Deniel, Y. On the a.s. Cesàro-α convergence for stationary or orthogonal random variables. J Theor Probab 2, 475–485 (1989). https://doi.org/10.1007/BF01051879
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DOI: https://doi.org/10.1007/BF01051879