Journal of Probability and Statistics

Journal of Probability and Statistics / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 873274 | 7 pages | https://doi.org/10.1155/2009/873274

A Note on Strong Convergence of Sums of Dependent Random Variables

Academic Editor: Mohammad Fraiwan Al-Saleh
Received05 Aug 2009
Revised25 Nov 2009
Accepted03 Dec 2009
Published22 Dec 2009

Abstract

For a sequence of dependent square-integrable random variables and a sequence of positive constants , conditions are provided under which the series converges almost surely as . These conditions are weaker than those provided by Hu et al. (2008).

1. Introduction and Results

Let be a sequence of square-integrable random variables defined on a probability space and let be a sequence of positive constants. The random variables are not assumed to be independent. Past research has focussed on conditions that ensure the strong convergence of two distinct but related series: If the second sequence converges to 0 almost surely, then is said to obey the strong law of large numbers (SLLN).

Assume that there exists a sequence of constants such that Our interest is in conditions on the growth rates of , , and which imply strong convergence of the above series.

There is an extensive literature on strong laws for independent random variables. Strong laws have been derived for various dependence structures such as negative association (e.g., Kuczmaszewska [1]), quasi-stationarity (e.g., Móricz [2], Chobanyan et al. [3]), and orthogonality (e.g., Stout [4]).

Hu et al. [5] focus on the strong convergence of the series without imposing strong conditions on the nature of the variances and covariances. Our aim is to weaken their condition on the covariances and establish the following theorem.

Theorem 1.1. Let be a sequence of square-integrable random variables and suppose that there exists a sequence of constants such that (1.2) holds. Let be a sequence of positive constants. Assume that there exists a constant such that, for all , Suppose that Then

To motivate the general nature of our result consider the following example. Let be a sequence of zero mean random variables where where is a stationary time series with autocovariance function and is a sequence of independent, zero mean random variables distributed independently of . Let Var Thus what we observe is an underlying stationary series disturbed by a noise process with variance that can depend on

We have Var and Cov, Condition (3.1) in Theorem of Hu et al. [5], which is the same as (1.4), is a constraint on the values whereas their condition (3.2) is a constraint on . In Chapter 2 of Stout [4] the condition on the variances is shown to be close to optimal for sequences of orthogonal random variables. Lyons [6] provides an SLLN for random variables with bounded variances under the condition One might conjecture that the condition (1.8) could be relaxed to The above theorem, whilst allowing for far more general models than (1.7), moves us closer to this constraint on the values.

For long range dependent stationary processes we have where and is a slowly varying function. Theorem 1.1 enables the strong convergence result to be extended to processes where the correlation decays at a slower rate than for

Applying Kronecker's lemma the strong law of large numbers result is an immediate consequence of the above theorem.

Corollary 1.2. Under the conditions of Theorem 1.1, if is monotone increasing, the strong law of large numbers holds, that is,

There are strong law results under weaker conditions than (1.5) but with stronger conditions on the variance (see, e.g., Lyons [6], Chobanyan et al. [3]). Both papers show that if the summands have bounded variance, then (1.5) can be weakened to Our approach focusses on the convergence of the series in (1.6) and relies on Kronecker's Lemma to obtain the strong law. If the aim is purely to obtain the SLLN, then alternative conditions might be possible as it is possible to construct sequences and such that but diverges. For example, take and Thus we can have the strong law holding but the series in (1.6) diverging.

2. Proofs

Throughout this paper, the symbol denotes a generic constant which is not necessarily the same at each appearance. We first prove a number of lemmas that enable us to obtain tighter bounds for key expressions in the proof of Theorem of Hu et al. [5].

Lemma 2.1. Let be a sequence of square-integrable random variables and suppose that there exists a sequence of constants such that (1.2) holds and a sequence satisfying (1.3). Then for all , ,

Proof. For all , ,

Lemma 2.2. For ,

Proof. Note that is an increasing function for Thus, for , Hence for ,

Lemma 2.3. For define Then , and, in general,

Proof. The result for is the sum of a standard geometric progression. The general result follows by noting Thus

Proof of Theorem 1.1. We will follow the method of proof in Theorem in Hu et al. [5]. To prove (1.6) we first show that is a Cauchy sequence for convergence in which will imply convergence in probability. Using Lemmas 2.1 and 2.2, Therefore there exists a random variable such that
Next we will show that a.s. Let be arbitrary. Note where the last line follows by using (1.4) and (1.5). Thus by the Borel Cantelli lemma almost surely. To finish the proof we utilize the generalization of the Rademacher-Menchoff maximal inequality given by Serfling [7] and argue as in Hu et al. [5]. It is sufficient to show that, for any , Using Serfling's inequality and (3.8) from Hu et al. [5]

References

  1. A. Kuczmaszewska, “The strong law of large numbers for dependent random variables,” Statistics & Probability Letters, vol. 73, no. 3, pp. 305–314, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. F. Móricz, “The strong laws of large numbers for quasi-stationary sequences,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 38, no. 3, pp. 223–236, 1977. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  3. S. Chobanyan, S. Levental, and H. Salehi, “Strong law of large numbers under a general moment condition,” Electronic Communications in Probability, vol. 10, pp. 218–222, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  4. W. F. Stout, Almost Sure Convergence, Academic Press, New York, NY, USA, 1974. View at: MathSciNet
  5. T.-C. Hu, A. Rosalsky, and A. I. Volodin, “On convergence properties of sums of dependent random variables under second moment and covariance restrictions,” Statistics & Probability Letters, vol. 78, no. 14, pp. 1999–2005, 2008. View at: Publisher Site | Google Scholar | MathSciNet
  6. R. Lyons, “Strong laws of large numbers for weakly correlated random variables,” The Michigan Mathematical Journal, vol. 35, no. 3, pp. 353–359, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. R. J. Serfling, “Moment inequalities for the maximum cumulative sum,” Annals of Mathematical Statistics, vol. 41, pp. 1227–1234, 1970. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2009 Tien-Chung Hu and Neville C. Weber. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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