Cesàro Summation for Random Fields

Journal of Theoretical Probability (Impact Factor: 0.86). 05/2009; 23(3):715-728. DOI: 10.1007/s10959-009-0223-9
Source: arXiv


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

Download full-text


Available from: Ulrich Stadtmueller
  • Source
    • "Strong limit theorems for random fields where the finiteness is a consequence of the moment assumption. As for the second step, this is a technically pretty involved matter for which we refer to Gut et al. (2010). For the analysis of T n k ,n k +an k we use the Kolmogorov upper exponential bounds (see e.g., Gut (2007), Lemma 8.2.1) and obtain (after having taken care of the centering inflicted by the truncation), "
    [Show abstract] [Hide abstract]
    ABSTRACT: The aim of the present paper is to review some joint work with Ulrich Stadtmüller concerning random field analogs of the classical strong laws. In the first part, we start, as background information, by quoting the law of large numbers and the law of the iterated logarithm for random sequences as well as for random fields, and the law of the single logarithm (LSL) for sequences. We close with a one-dimensional LSL pertaining to windows, whose edges expand in an “almost linear fashion”, viz., the length of the nth window equals, for example, n/logn or n/loglogn. A sketch of the proof is also given. The second part contains some extensions of the LSL to random fields, after which we turn to convergence rates in the law of large numbers. Departing from the now legendary Baum-Katz theorem in 1965, we review a number of results in the multiindex setting. Throughout the text, the main emphasis is on the case of “non-equal expansion rates”, viz., the case when the edges along the different directions expand at different rates. Some results when the power weights are replaced by almost exponential weights are also given. We close with some remarks on martingales and the strong law.
    Preview · Article · Jan 2012
  • Source
    • "We wish to generalize the multiindex version so that instead of normalizing with |n| 1/r we normalize with different powers for different coordinates. Looking into Cesàro means for random fields (see [4]) it is of interest to control these sums. The main results are presented in the following section followed by a section with proofs. "
    [Show abstract] [Hide abstract]
    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.
    Full-text · Article · Apr 2009 · Statistics [?] Probability Letters