Conditional acceptability of random variables
- Tasos C Christofides^{1}Email author,
- István Fazekas^{2} and
- Milto Hadjikyriakou^{3}
https://doi.org/10.1186/s13660-016-1093-1
© Christofides et al. 2016
Received: 23 December 2015
Accepted: 25 May 2016
Published: 6 June 2016
Abstract
Acceptable random variables introduced by Giuliano Antonini et al. (J. Math. Anal. Appl. 338:1188-1203, 2008) form a class of dependent random variables that contains negatively dependent random variables as a particular case. The concept of acceptability has been studied by authors under various versions of the definition, such as extended acceptability or wide acceptability. In this paper, we combine the concept of acceptability with the concept of conditioning, which has been the subject of current research activity. For conditionally acceptable random variables, we provide a number of probability inequalities that can be used to obtain asymptotic results.
Keywords
MSC
1 Introduction
Let \(\{X_{n},n\in\mathbb{N}\}\) be a sequence of random variables defined on a probability space \((\Omega, \mathcal{A},\mathcal{P})\). Giuliano Antonini et al. [1] introduced the concept of acceptable random variables as follows.
Definition 1
The class of acceptable random variables includes in a trivial way collections of independent random variables. But in most cases, acceptable random variables exhibit a form of negative dependence. In fact, as Giuliano Antonini et al. [1] point out, negatively associated random variables with a finite Laplace transform satisfy the notion of acceptability. However, acceptable random variables do not have to be negatively dependent. A classical example of acceptable random variables that are not negatively dependent can be constructed based on problem III.1 listed in the classical book of Feller [2]. Details can be found in Giuliano Antonini et al. [1], Shen et al. [3], or Sung et al. [4].
The idea of acceptability has been modified or generalized in certain directions. For example, Giuliano Antonini et al. [1] introduced the concept of m-acceptable random variables, whereas other authors provided weaker versions such as the notions of extended acceptability or wide acceptability. The following definition given by Sung et al. [4] provides a weaker version of acceptability by imposing a condition on λ.
Definition 2
The concept of acceptable random variables has been studied extensively by a few authors, and a number of results are available in the literature such as exponential inequalities and complete convergence results. For the interested reader, we suggest the papers of Giuliano Antonini et al. [1], Sung et al. [4], Wang et al. [5], Shen et al. [3], among others.
Further, in addition to the definition of acceptability, Choi and Baek [6] introduced the concept of extended acceptability as follows.
Definition 3
It is clear that acceptable random variables are extended acceptable. The following example provides a sequence of random variables that satisfies the notion of extended acceptability.
Example 4
Observe that \(\{X_{n},n\in\mathbb{N}\}\) is a strictly stationary sequence that is negatively dependent for \(\beta<0\) and positively dependent for \(\beta>0\).
For the class of extended acceptable random variables, Choi and Baek [6] provide an exponential inequality that enables the derivation of asymptotic results based on complete convergence.
A different version of acceptability, the notion of wide acceptability, is provided by Wang et al. [5].
Definition 5
The following example gives random variables that satisfy the definition of wide acceptability.
Example 6
The concept of widely acceptable random variables follows naturally from the concept of wide dependence of random variables introduced by Wang et al. [7]. Wang et al. [8] and Wang et al. [7] stated (without proof) that, for widely orthant dependent random variables, the inequality in Definition 5 is true for any λ. For widely acceptable random variables, Wang et al. [5] pointed out, although did not provide the details, that one can get exponential inequalities similar to those obtained for acceptable random variables.
In this paper, we combine the concept of conditioning on a σ-algebra with the concept of acceptability (in fact, wide acceptability) and define conditionally acceptable random variables. In Section 2.1, we give the basic definitions and examples and prove some classical exponential inequalities. In Section 2.2, we provide asymptotic results by utilizing the tools of Section 2.1. Finally, in Section 3, we give some concluding remarks.
2 Results and discussion
Recently, various researchers have studied extensively the concepts of conditional independence and conditional association (see, e.g., Chow and Teicher [9], Majerak et al. [10], Roussas [11], and Prakasa Rao [12]) providing conditional versions of known results such as the generalized Borel-Cantelli lemma, the generalized Kolmogorov inequality, and the generalized Hájek-Rényi inequalities. Counterexamples are available in the literature, proving that the conditional independence and conditional association are not equivalent to the corresponding unconditional concepts.
The concept of conditional negative association was introduced by Roussas [11]. Let us recall its definition since it is related to the results presented further.
Definition 7
As mentioned earlier, it can be shown that the concepts of negative association and conditional negative association are not equivalent. See, for example, Yuan et al. [13], where various of counterexamples are given.
2.1 Conditional acceptability
In this paper, we define the concept of conditional acceptability by combining the concept of conditioning on a σ-algebra and the concept of acceptability. In particular, conditioning is combined with the concept of wide acceptability. We therefore give the following definition.
Definition 8
Remark 9
Remark 10
It can be easily verified that if random variables \(X_{1},\ldots,X_{n}\) are \(\mathcal{F}\)-acceptable, then the random variables \(X_{1}-E^{\mathcal {F}}(X_{1}), X_{2}-E^{\mathcal{F}}(X_{2}),\ldots,X_{n}-E^{\mathcal{F}}(X_{n})\) are also \(\mathcal{F}\)-acceptable, and \(- X_{1} , - X_{2} , \ldots, - X_{n}\) are also \(\mathcal{F}\)-acceptable.
The random variables given in the following example satisfy the definition of \(\mathcal{F}\)-acceptability.
Example 11
In the case where \(\mathcal{F}\) is chosen to be the trivial σ-algebra, that is, \(\mathcal{F} = \{\emptyset,\Omega\}\), the definition of \(\mathcal{F}\)-acceptability reduces to the definition of unconditional wide acceptability. The converse statement cannot always be true, and this can be proven via the following counterexample, showing that the concepts of \(\mathcal{F}\)-acceptability and acceptability are not equivalent.
Example 12
It is well known that exponential inequalities played an important role in obtaining asymptotic results for sums of independent random variables. Classical exponential inequalities were obtained, for example, by Bernstein, Hoeffding, Kolmogorov, Fuk, and Nagaev (see the monograph of Petrov [14]). A crucial step in proving an exponential inequality is the use of an inequality like that in Definition 2. Next, we provide several exponential inequalities for \(\mathcal{F}\)-acceptable random variables.
The following Hoeffding-type inequality is obtained by Yuan and Xie [16].
Lemma 13
The result that follows is a conditional version of the well-known Hoeffding inequality (Hoeffding [15], Theorem 2). Similar results were proven by Shen et al. [3], Theorem 2.3, for acceptable random variables and by Yuan and Xie [16], Theorem 1, for conditionally linearly negatively quadrant dependent random variables. Our result improves Theorem 1 of Yuan and Xie [16].
Theorem 14
Proof
The result that follows is the conditional version of Theorem 2.1 of Shen et al. [3].
Theorem 15
Proof
Theorem 16
Proof
The probability inequalities presented above were proven under the assumption of bounded random variables. The result that follows provides a probability inequality under a moment condition.
Theorem 17
Proof
Remark 18
Since \(\{X_{n}, n\in\mathbb{N}\}\) is a sequence of \(\mathcal {F}\)-acceptable random variables with \(g(n) \equiv1\), where \(\mathcal {F}\) is the trivial σ-algebra, the above result is reduced to the result of Corollary 2.1 of Shen and Wu [17].
2.2 Conditional complete convergence
Complete convergence results are well known for independent random variables (see, e.g., Gut [18]). The classical results of Hsu, Robbins, Erdős, Baum, and Katz were extended to certain dependent sequences. Using the results of Section 2.1, we can show the complete convergence for the partial sum of \(\mathcal{F}\)-acceptable random variables under various assumptions. We will need the following definition of conditional complete convergence (see Christofides and Hadjikyriakou [19] for details).
Definition 19
Theorem 20
Proof
Remark 21
Next, we provide a result, which is a conditional version of Theorem 3.2 of Shen et al. [3].
Theorem 22
Proof
Theorem 23
Proof
The theorem that follows gives a conditional exponential inequality for the partial sum of \(\mathcal{F}\)-acceptable random variables, under a moment condition, which, in the unconditional case is a condition appearing very frequently in large deviation results (see, e.g., Nagaev [20] and Teicher [21]). It also appears as condition (3.3) of Theorem 3.3 of Shen et al. [3]. However, the bound provided here allows us to prove the complete convergence, and in the unconditional case, under assumptions different from those of Theorem 3.3 of Shen et al. [3].
Theorem 24
- (i)If \(\frac{1}{H} [ 1 - \sqrt{\frac{B_{n}^{2}}{2Hx + B_{n}^{2}}} ]<\delta\), then for an a.s. positive \(\mathcal{F}\)-measurable random variable x, we have$$P^{\mathcal{F}} \Biggl( \Biggl\vert \sum_{i=1}^{n}X_{i} \Biggr\vert \geq x \Biggr) \leq2g(n) \exp \biggl[ - \frac{1}{2 H^{2}} { \Bigl( \sqrt{2Hx + B_{n}^{2}} - \sqrt{B_{n}^{2}} \Bigr)}^{2} \biggr] \quad \textit{a.s.} $$
- (ii)If \(g(n)\leq K\) a.s. for all n, where K is a.s. finite and \(\{ B_{n}^{2}\}\in\mathcal{H}\) a.s., then$$\frac{S_{n}}{B_{n}^{2}}\textit{ converges completely to }0\textit{ given } \mathcal{F}. $$
Proof
Remark 25
In the previous theorem, it is assumed that \(g(n) \leq K\) a.s. for every n, where K is finite a.s. However, we may have the complete convergence without this assumption. For example, the RHS of (10) may be finite even when g is not bounded. Similar statements can be made for Theorems 20 and 22.
3 Conclusions
In this paper, we define the class of conditionally acceptable random variables as a generalization of the class of acceptable random variables studied previously by Giuliano Antonini et al. [1], Shen et al. [3] and Sung et al. [4], among others. The idea of conditioning on a σ-algebra is gaining increasing popularity with potential applications in fields such as risk theory and actuarial science. For the class of conditionally acceptable random variables, we provide useful probability inequalities, mainly of the exponential type, which can be used to establish asymptotic results and, in particular, complete convergence results. We anticipate that the results presented in this paper will serve as a basis for research activity, which will yield further theoretical results and applications.
Declarations
Acknowledgements
The authors are grateful to the two anonymous referees for their valuable comments, which led to a much improved version of the manuscript.
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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