Cesàro Summation for Random Fields

Journal of Theoretical Probability (Impact Factor: 0.79). 05/2009; 23(3):715-728. DOI: 10.1007/s10959-009-0223-9
Source: arXiv

ABSTRACT Various methods of summation for divergent series of real numbers have been generalized to analogous results for sums of i.i.d.
random variables. The natural extension of results corresponding to Cesàro summation amounts to proving almost sure convergence
of the Cesàro means. In the present paper we extend such results as well as weak laws and results on complete convergence
to random fields, more specifically to random variables indexed by ℤ+2, the positive two-dimensional integer lattice points.

KeywordsCesàro summation-Sums of i.i.d. random variables-Complete convergence-Convergence in probability-Almost sure convergence-Strong law of large numbers
Mathematics Subject Classification (2000)60F15-60G50-60G60-40G05-60F05

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Available from: Ulrich Stadtmueller, Jul 06, 2015
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    ABSTRACT: The classical Marcinkiewicz-Zygmund law for i.i.d. random variables has been generalized by Gut [Gut, A., 1978. Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann. Probab. 6, 469-482] to random fields. Therein all indices have the same power in the normalization. Looking into some weighted means of random fields, such as Cesro summation, it is of interest to generalize these laws to the case where different indices have different powers in the normalization. In this paper we give precise moment conditions for such laws.
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