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Some mean convergence theorems for arrays of rowwise pairwise negative quadrant dependent random variables
- Tapas K. Chandra^1,
- Deli Li2 and
- Andrew Rosalsky3Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-018-1811-y
© The Author(s) 2018
- Received: 9 May 2018
- Accepted: 26 July 2018
- Published: 23 August 2018
Abstract
For arrays of rowwise pairwise negative quadrant dependent random variables, conditions are provided under which weighted averages converge in mean to 0 thereby extending a result of Chandra, and conditions are also provided under which normed and centered row sums converge in mean to 0. These results are new even if the random variables in each row of the array are independent. Examples are provided showing (i) that the results can fail if the rowwise pairwise negative quadrant dependent hypotheses are dispensed with, and (ii) that almost sure convergence does not necessarily hold.
Keywords
- Array of rowwise pairwise negative quadrant dependent random variables
- Pairwise independent random variables
- Weighted averages
- Degenerate mean convergence
- Stochastic domination
- Almost sure convergence
MSC
- 60F25
- 60F99
- 60F05
1 Introduction
Theorem 1.1
(Chandra [3], Theorem 1)
- (i)
Our results pertain to weighted averages either from an array of random variables whose nth row is comprised of \(k_{n}\) pairwise negative quadrant dependent random variables, \(n \geq 1\) (Theorem 3.1) or from an array of random variables whose nth row is comprised of \(k_{n}\) pairwise independent random variables, \(n \geq 1\) (Theorem 3.2). No independence or dependence conditions are imposed between the random variables from different rows of the arrays. The Chandra [3] result considered weighted averages from a sequence of pairwise i.i.d. random variables.
- (ii)
The random variables that we consider are assumed to be stochastically dominated by a random variable which is a weaker assumption than the assumption of Chandra [3] that the random variables are identically distributed.
The third main result (Theorem 3.3) establishes for an array of random variables whose nth row is comprised of \(k_{n}\) pairwise negative quandrant dependent random variables, \(n \geq 1\) a degenerate mean convergence result for normed and centered row sums. In contradistinction to Theorems 3.1 and 3.2, weighted averages and stochastic domination play no role in Theorem 3.3. As in Theorems 3.1 and 3.2, no independence or dependence conditions are imposed between the random variables from different rows of the array in Theorem 3.3.
Definition 1.1
It is of course immediate that if \(X_{1}, \ldots, X_{N} \) are pairwise independent (a fortiori, independent) random variables, then \(\{X_{1}, \ldots, X_{N} \}\) is PNQD.
In many stochastic models, the classical assumption of independence among the random variables in the model is not a reasonable one; the random variable may be “repelling” in the sense that small values of any of the random variables increase the probability that the other random variables are large. Thus an assumption of some type of negative dependence is often more suitable. Pemantle [11] prepared an excellent survey on a general “theory of negative dependence”.
A collection of N PNQD random variables arises by sampling without replacement from a set of \(N \geq 2\) real numbers (see, e.g., Bozorgnia et al. [2]). Li et al. [7] showed that for every set of \(N \geq 2\) continuous distribution functions \(\{F_{1}, \ldots, F_{N} \}\), there exists a set of PNQD random variables \(\{X_{1}, \ldots, X_{N} \}\) such that the distribution function of \(X_{j}\) is \(F_{j}\), \(1 \leq j \leq N\) and such that for all \(j \in \{1, \ldots, N-1 \}\), \(X_{j}\) and \(X_{j+1}\) are not independent.
An array of random variables \(\{X_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \}\) is said to be rowwise PNQD if for each \(n \geq 1\), the set of random variables \(\{X_{n,j}, 1 \leq j \leq k_{n} \}\) is PNQD. There is interesting literature of investigation on the strong law of large numbers problem for row sums of rowwise PNQD arrays; see the discussion in Li et al. [7].
Definition 1.2
Remark 1.1
Condition (1.3) is, of course, automatic with \(X = X_{1,1}\) and \(D = 1\) if the array \(\{X_{n,j}, 1 \leq j \leq k _{n}, n \geq 1 \}\) consists of identically distributed random variables.
2 Preliminary lemmas
Three lemmas will now be stated. Lemmas 2.1, 2.2, and 2.3 are used in the proof of Theorem 3.1, Lemma 2.3 is used in the proof of Theorem 3.2, and Lemmas 2.1 and 2.2 are used in the proof of Theorem 3.3.
Lemma 2.1 follows from Lemma 1 of Lehmann [6]; see Matuła [8] for a more direct proof.
Lemma 2.1
Let the set of random variables \(\{ X_{1}, \ldots, X_{N} \} \) be PNQD, and for each \(j \in \{1, \ldots, N \}\), let \(f_{j}: \mathbb{R} \rightarrow \mathbb{R}\). If the functions \(f_{1}, \ldots, f_{N}\) are all nondecreasing or all nonincreasing, then the set of random variables \(\{ f_{1}(X _{1}), \ldots, f_{N}(X_{N}) \} \) is PNQD.
The next lemma is well known (see, e.g., Patterson and Taylor [10]).
Lemma 2.2
The following lemma is essentially due to Adler et al. [1].
Lemma 2.3
(Adler et al. [1])
3 Mainstream
The main results, Theorems 3.1–3.3, may now be established. These are new results even under the stronger hypothesis that the random variables in each row of the array are independent.
Theorem 3.1
Proof
Remark 3.1
The next theorem is a version of Theorem 3.1 without assumption (3.1) for an array of random variables where the random variables in each row of the array are pairwise independent (which is a stronger assumption than the array being rowwise PNQD).
Theorem 3.2
Proof
Remark 3.2
The cited result of Chandra [3] follows immediately from Theorem 3.2 by taking \(k_{n} = n\), \(n \geq 1\) and \(X_{n,j} = X_{j}\), \(1 \leq j \leq n\), \(n \geq 1\).
Remark 3.3
We now show via an example that the hypotheses to Theorems 3.1 and 3.2 do not necessarily ensure that \(\sum_{j=1}^{k_{n}} a_{n,j}X_{n,j} \longrightarrow 0\) almost surely (a.s.).
Example 3.1
We now establish Theorem 3.3. Throughout the rest of this section, for an array of random variables \(\{ X_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \} \), let \(S_{n} = \sum_{j=1}^{k_{n}} X_{n,j}\), \(n \geq 1\).
Theorem 3.3
Proof
Corollary 3.1
Proof
Remark 3.4
Example 3.2
Corollary 3.2
Proof
4 Conclusions
- (i)where \(\mathbb{E}X_{n,j} = 0\), \(1 \leq j \leq k_{n}\), \(n \geq 1\), and \(\{ a_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \} \) is an array of constants;$$\sum_{j=1}^{k_{n}}a_{n,j}X_{n,j} \stackrel{\mathscr{L}_{1}}{\longrightarrow} 0, $$
- (ii)where \(\{d_{n}, n \geq 1 \}\) is a sequence of positive constants.$$\frac{\sum_{j=1}^{k_{n}} ( X_{n,j} - \mathbb{E}X_{n,j} ) }{d_{n}} \stackrel{\mathscr{L}_{1}}{\longrightarrow } 0, $$
A version of the result in (i) is also obtained for an array of rowwise pairwise independent random variables and this result extends the result of Chandra [3]. Examples are provided showing that the above results can fail if the hypotheses are weakened and that a.s. convergence does not necessarily hold together with the \(\mathscr{L}_{1}\) convergence.
Declarations
Acknowledgements
The authors are grateful to the reviewers for carefully reading the manuscript and for offering helpful comments and suggestions.
Authors’ information
Tapas K. Chandra deceased before publication of this work was completed.
Funding
The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (Grant #: RGPIN-2014-05428).
Authors’ contributions
All authors contributed equally and significantly in writing this article. All the authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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