# Conditional acceptability of random variables

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## 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

\(\mathcal{F}\)-acceptable random variables conditional complete convergence exponential inequalities## MSC

60F15 62G20## 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

*λ*,

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

*λ*,

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

*C*can be obtained by the formula

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.

*σ*-algebra of \(\mathcal{A}\). Furthermore, \(\operatorname{Cov}^{\mathcal {F}}(X,Y)\) denotes the conditional covariance of

*X*and

*Y*given \(\mathcal{F}\), where

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

*A*and

*B*of \({1, 2, \ldots, n}\) and for any real-valued componentwise nondecreasing functions

*f*,

*g*on \(\mathbb{R}^{|A|}\) and \(\mathbb{R}^{|B|}\), respectively, where \(|A| = \operatorname{card}(A)\), provided that the covariance is defined. An infinite collection is conditionally negatively associated given \(\mathcal{F}\) if every finite subcollection is \(\mathcal{F}\)-NA.

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

*i*and, for any \(|\lambda|< \delta\), there exist positive numbers \(g(n)\), \(n\geq1\), such that

### Remark 9

*λ*to be a random object. Thus, as a result, if the random variables \(X_{1}, X_{2},\ldots,X_{n}\) are \(\mathcal{F}\)-acceptable for \(\delta >0\), then

*λ*such that \(|\lambda|<\delta\) a.s.

### 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

*σ*-algebra generated by

*B*. Yuan

*et al.*[13] proved that \(\{X_{1},X_{2}\}\) are \(\mathcal{F}\)-NA. By proposition P1 of the same paper it follows that, for all \(\lambda \in\mathbb{R}\), \(\{e^{\lambda X_{1}},e^{\lambda X_{2}}\}\) are \(\mathcal{F}\)-NA, and therefore \(\{X_{1},X_{2}\}\) are \(\mathcal {F}\)-acceptable for \(g(2) =1\).

*X*be a random variable, \(X\geq0\) a.s., and let

*ϵ*be an \(\mathcal{F}\)-measurable random variable such that \(\epsilon>0\) a.s. It is known that

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

*Assume that*\(P(a\leq X\leq b) = 1\),

*where*

*a*

*and*

*b*

*are*\(\mathcal{F}\)-

*measurable random variables such that*\(a< b\)

*a*.

*s*.

*Then*

*for any*\(\mathcal{F}\)-

*measurable random variable*

*λ*.

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

*Let*\(X_{1}, X_{2},\ldots,X_{n}\)

*be*\(\mathcal{F}\)-

*acceptable random variables for*\(\delta>0\)

*such that*\(P(a_{i}\leq X_{i}\leq b_{i})=1\), \(i=1,2,\ldots,n\),

*where*\(a_{i}\)

*and*\(b_{i}\)

*are*\(\mathcal {F}\)-

*measurable random variables such that*\(a_{i}< b_{i}\)

*a*.

*s*.

*for all*

*i*.

*Then for an*\(\mathcal{F}\)-

*measurable*

*ϵ*

*with*\(0<\epsilon<\frac {\delta}{4}\sum_{i=1}^{n}(b_{i}-a_{i})^{2}\)

*a*.

*s*.,

*we have*

*where*\(S_{n} = \sum_{i=1}^{n}X_{i}\).

### Proof

*λ*be an \(\mathcal{F}\)-measurable random variable such that \(0<\lambda<\delta\) a.s. Then by the conditional version of Markov’s inequality

The result that follows is the conditional version of Theorem 2.1 of Shen *et al.* [3].

### Theorem 15

*Let*\(X_{1},\ldots,X_{n}\)

*be*\(\mathcal{F}\)-

*acceptable random variables for*\(\delta>0\).

*Assume that*\(E^{\mathcal{F}}(X_{i})=0\)

*and that*\(E^{\mathcal {F}}(X_{i}^{2})\)

*is finite for any*

*i*.

*Let*\(B_{n}^{2} = \sum_{i=1}^{n}E^{\mathcal{F}}(X_{i}^{2})\).

*Assume that*\(|X_{i}|\leq cB_{n}\)

*a*.

*s*.,

*where*

*c*

*is a positive*\(\mathcal{F}\)-

*measurable random variable*.

*Then*

*for any positive*\(\mathcal{F}\)-

*measurable random variable*

*ϵ*.

### Proof

*t*be a positive \(\mathcal{F}\)-measurable random variable such that \(tcB_{n}\leq1\). Note that, for \(k\geq2\),

*ϵ*, \(B_{n}\), and

*c*are positive \(\mathcal{F}\)-measurable random variables, the conditional Markov inequality can be applied, and by using the above calculations we have that, for \(0< t<\delta\),

### Theorem 16

*Let*\(X_{1},\ldots,X_{n}\)

*be*\(\mathcal{F}\)-

*acceptable random variables for*\(\delta>0\).

*Assume that*\(E^{\mathcal{F}}(X_{i}) = 0\)

*a*.

*s*.

*and that there is an*\(\mathcal{F}\)-

*measurable random variable*

*b*

*such that*\(|X_{i}|\leq b\)

*a*.

*s*.

*for all*

*i*.

*Define*\(B_{n}^{2}=\sum_{i=1}^{n}E^{\mathcal{F}} (X_{i}^{2} )\).

*Then*,

*for any positive*\(\mathcal{F}\)-

*measurable random variable*

*ϵ*

*with*\(\frac {\epsilon}{B_{n}^{2}+\frac{b}{3}\epsilon}<\delta\),

*we have*

*and*

### 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

*Let*\(\{X_{n},n\in\mathbb{N}\}\)

*be a sequence of*\(\mathcal{F}\)-

*acceptable random variables for*\(\delta>0\),

*and let*\(\{ c_{n},n\in\mathbb{N}\}\)

*be a sequence of positive*\(\mathcal {F}\)-

*measurable random variables with*\(C_{n} = \sum_{i=1}^{n}c_{i}\)

*for all*\(n=1,2,\ldots\) .

*Assume that there is a positive*\(\mathcal {F}\)-

*measurable random variable*

*T*

*with*\(T\leq\delta\)

*a*.

*s*.

*such that*,

*for*\(|t|\leq T\)

*and fixed*\(n\geq1\),

*Then*,

*for any positive*\(\mathcal{F}\)-

*measurable random variable*

*ϵ*,

### Proof

*T*and \(\{c_{n},n\in\mathbb{N}\}\). So \(-X_{1},-X_{2},\ldots,-X_{n}\) are \(\mathcal{F}\)-acceptable random variables that satisfy condition (4) for \(0\leq t\leq T\), and therefore by applying inequalities (5) and (6) for \(-S_{n}\) we have that

### 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

*X*if

*ϵ*such that \(\epsilon>0\) a.s.

*et al.*[3]:

### Theorem 20

*Let*\(X_{1}, X_{2},\ldots\)

*be a sequence of*\(\mathcal {F}\)-

*acceptable random variables for*\(\delta>0\).

*Assume that*\(E^{\mathcal{F}}(X_{i}) = 0\)

*a*.

*s*.

*and*\(|X_{i}|\leq b\)

*a*.

*s*.

*for all*

*i*,

*where*

*b*

*is an*\(\mathcal{F}\)-

*measurable random variable*.

*Assume that*\(g(n)< K\)

*a*.

*s*.,

*where*

*K*

*is an a*.

*s*.

*finite*\(\mathcal{F}\)-

*measurable random variable*.

*Let*\(B_{n}^{2} = \sum_{i=1}^{n}E^{\mathcal{F}} ( X_{i}^{2} )\)

*and assume that*\(\{B_{n}^{2},n\in\mathbb{N}\}\in\mathcal {H}\)

*a*.

*s*.

*Then*

### Proof

### Remark 21

*et al.*[3].

Next, we provide a result, which is a conditional version of Theorem 3.2 of Shen *et al.* [3].

### Theorem 22

*Let*\(X_{1},X_{2},\ldots\)

*be a sequence of*\(\mathcal{F}\)-

*acceptable random variables for*\(\delta> 0\).

*Assume that*\(|X_{i}|\leq c\)

*a*.

*s*.,

*where*

*c*

*is an*\(\mathcal{F}\)-

*measurable random variable such that*\(c>0\)

*a*.

*s*.

*Assume that*\(g(n)\leq K\)

*a*.

*s*.,

*where*

*K*

*is an a*.

*s*.

*finite*\(\mathcal {F}\)-

*measurable random variable*.

*Moreover*,

*let*\({b_{n}}\in\mathcal{H}\)

*a*.

*s*.,

*where*\(b_{n}\)

*are*\(\mathcal{F}\)-

*measurable*.

*Then*

### Proof

### Theorem 23

*Let*\(X_{1},X_{2},\ldots\)

*be a sequence of*\(\mathcal{F}\)-

*acceptable random variables for*\(\delta> 0\)

*such that all the assumptions of Theorem*17

*are satisfied*.

*If*,

*for any*\(\epsilon>0\)

*a*.

*s*.,

*where*\(\{b_{n},n\in\mathbb{N}\}\)

*is a sequence of positive*\(\mathcal {F}\)-

*measurable random variables*,

*then*

### 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

*Let*\(X_{1},X_{2},\ldots\)

*be a sequence of*\(\mathcal{F}\)-

*acceptable random variables for*\(\delta>0\).

*Assume that*\(E^{\mathcal{F}}(X_{i}) = 0\)

*and let*\(\sigma_{i}^{2} = E^{\mathcal{F}}(X_{i}^{2})\)

*be a*.

*s*.

*finite*.

*Let*\(B_{n}^{2} = \sum_{i=1}^{n}\sigma_{i}^{2}\).

*Assume that there exists an a*.

*s*.

*positive and a*.

*s*.

*finite*\(\mathcal{F}\)-

*measurable random variable*

*H*

*such that*

- (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

*t*is an \(\mathcal{F}\)-measurable random variable. If \(|t|\leq\frac{1}{H}\), then

### 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.

## Notes

### Acknowledgements

The authors are grateful to the two anonymous referees for their valuable comments, which led to a much improved version of the manuscript.

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