Abstract
In this paper, we investigate the second Borel–Cantelli lemma for capacity without the assumption of independence for events. We obtain a sufficient condition under which the second Borel–Cantelli lemma for capacity holds. Our results are natural extensions of the classical Borel–Cantelli lemma. However, the proof is different from the existing literature.
Keywords:
capacity; Choquet expectation; exponential independence; law of large numbers; sub-linear expectation MSC:
60A10; 60H05
1. Introduction
In probability theory, the Borel–Cantelli lemma, which was first obtained by Borel (1909, 1912) [,] and Cantelli (1917) [], is an important theorem about sequences of events, such a lemma consists of two parts, which are called the first and second Borel–Cantelli lemma. The lemma states that, under certain conditions, an event will occur with either probability zero or probability one. The second Borel–Cantelli lemma is a partial converse of the first Borel–Cantelli lemma since the second Borel–Cantelli lemma needs the additional assumption of independence.
Since then, there has been a large amount of literature to extend the Borel–Cantelli lemma (see, for example, Chandra (2012) [] in detail). For the first Borel–Cantelli lemma, we refer the reader to Barndorff-Nielsen (1961) [], Martikainen and Petrov (1990) [], and Balakrishnan and Stepanov (2010) []. For the second Borel–Cantelli lemma, we refer the reader to Erdös and Rényi (1959) [], Kochen and Stone (1964) [], Petrov (2002, 2004) [,], Yan (2006) [], (which provides a new and simple proof about Kochen and Stone (1964) []), and Xie (2009) [] and Zong et al. (2016) [], in which many attempts can be founded to weaken the independence condition.
Recently, motivated by mathematical finance and robust statistics, non-additive measures or non-linear expectations have already caught many scholars’ attention; see Huber (1973) [], Denneberg (1994) [], Wang and Klir (2009) [], Peng (2006, 2009, 2019) [,,], Torra et al. (2014) [], and Zong et al. (2016) []. A natural question is whether or not such a Borel–Cantelli lemma could be extended to the case where the probability measure is non-additive.
According to the property of the first Borel–Cantelli lemma, we can easily extend the lemma to the case where the probability is no longer additive. In fact, the first Borel–Cantelli lemma holds for all set functions that are of countable subadditivity and monotonicity; see Billingsley (1995) []. However, for the second Borel–Cantelli lemma, it is not easy to do so because this lemma depends on the assumption of independence. Therefore, many concepts of independence under nonadditive probability/expectation have been introduced, for example, Peng’s independence in Peng (2006, 2009, 2019) [,,], Marinacci pre-independence in Maccheroni and Marinacci (2005) [], and Puhalskii independence (2001) [], as well as the Fubini-Like Theorem for Choquet Integrals, in Zong et al. (2016) []. A natural question is whether we could investigate the second Borel–Cantelli lemma for capacities without the assumption of independence. In this paper, we obtain a sufficient condition under which the second Borel–Cantelli lemma for capacity holds. It turns out that our results are natural extensions of the classical Borel–Cantelli lemma. However, the proofs are different from the existing literature.
This paper is organized as follows: In Section 2, we show some basic definitions and propositions with respect to capacity and present some preparatory lemmas. In Section 3, we provide a sufficient condition under which the second Borel–Cantelli lemma for capacities holds. In Section 4, we consider the case where random variables are independent under non-additive expectations or non-additive probabilities.
2. Preliminaries
Assume that is a measurable space; we define capacity, V, as follows.
Definition 1.
A set-function, V, on is called a capacity if it satisfies
- (i)
- (ii)
- (iii)
Given a capacity, V, let the -measurable function be a random variable defined on We focus on Choquet expectation, , denoted as
Definition 2.
Given a capacity, V, a Choquet (integral) expectation is denoted as
We assume that is the set of all random variables, X, with
Definition 3.
Two random variables, , are comonotone if, almost surely,
For more knowledge about comonotonicity, see, for instance, Dhaene et al. (2002) [].
The basic properties of Choquet expectations are given in the following proposition (see, e.g., Denneberg (1992) []).
Lemma 1.
- (a)
- Monotonicity: if , then .
- (b)
- Constant preserving: .
- (c)
- Translation invariance: .
- (d)
- Positive homogeneity: .
- (e)
- Lower–upper Choquet expectations:
- (f)
- Comonotonic additivity: if are comonotonic random variables, then
Remark 1.
Usually, a Choquet expectation does not satisfy the following sub-linearity:
However, it has been proven that a Choquet expectation satisfies sub-linearity if and only if the corresponding capacity, V, is —alternating in the sense of
Proposition 1.
Let φ on be a positively convex function. Then, the Jensen inequality under Choquet expectation holds:
Proof.
First, it is easy to check that a Choquet expectation has the following property:
In fact, obviously, the above inequality becomes equality if , by the definition of Choquet expectation. We now prove the case where In fact,
The last inequality is due to Lemma 1(e).
Using the above inequality, we can easily prove this lemma. Indeed, because the function is convex, there exists a countable set, D, in , such that Via the translation invariance of the Choquet expectation in Lemma 1(c), we have
The proof is complete. □
Because the probability measure in probability theory is assumed to be continuous in the sense that whenever , Fatou’s lemma is naturally true. However, for capacities, Fatou’s lemma is usually not true because the capacity in the nonlinear case is no longer continuous. Thus, we need the following concept.
Definition 4.
A capacity, V, is called a Fatou-like capacity if
It is easy to show that the following capacities are Fatou-like capacities:
Example 1.
Let and whenever and then V is a Fatou-like capacity.
Example 2.
Let be a weakly compact set of probability measures defined on ; then, the upper probability, V, defined by
is a Fatou-like capacity.
The following lemma is an important lemma that we use in this paper. The main idea is from Yan (2006) [].
Lemma 2.
Let X be a random variable, such that for a constant, and then
Proof.
It is easy to check that, for any
here and in the sequel, represents the indicator function of set
Therefore,
For convenience, we denote
It then follows the elementary inequality that
Taking the Choquet integration on both sides in inequality (3), according to the monotonicity of Choquet integration in Lemma 1(a), we have
Furthermore, it is easy to check that both and are co-monotonic; hence, via the comonotonic additivity of the Choquet expectation, we get
This, with (4), implies that
That is
which, with (2), implies the desired result. □
Lemma 3.
Suppose that is a sequence of events; set Assume that
and for any sequence of real numbers, with
Then,
- (I)
- For any constant, , and any sequence, with
- (II)
- For any
- (III)
- For any constant
Proof.
The proof of (I):
A trite calculation of a double integral, the following elementary equality can be obtained easily:
Immediately, we get
Choosing in (8).
Since and as thus, for a sufficiently large we have
Note the fact that Thus,
Set expectation on both sides of the inequality above; given the translation invariance in Lemma 1 and immediately,
The last inequality follows from the fact that, for
Therefore, for a sufficiently large we have
The proof of (I) is complete.
The proof of (II): For any via Markov’s inequality, we get
due to (I) and
The proof of (III): Let and . It is easy to obtain confirmation that and are comonotonic. Similarly to (3), we have
due to the comonotonic additivity of Choquet integration. Thereby, we obtain
due to (I) and (II). □
3. The Second Borel–Cantelli Lemma for Capacities
We now begin to prove the second Borel–Cantelli lemma for capacities:
Theorem 1.
Let V be a Fatou-like capacity, and let be a sequence of events, such that
If, for any sequence
then
Proof.
Set and . Immediately, , and
Now, we consider the following two events:
and here,
If holds, then is a finite number. Hence,
because of This implies Therefore, we have the following inclusion relation:
The monotonicity of the capacity V and the definition of Fatou-like capacity (1) imply that
In order to apply Lemma 2, we need to check the conditions of Lemma 2.
Given the assumption of Theorem that
and Jensen’s inequality, we have, for
Thanks to Lemma 2, we thus have
For the numerator in fraction (11), we see that
On the other hand, for the denominator in fraction (11), note that and we have
For the second term, for simplicity, we write ; via the comonotonic additivity of Choquet expectation, we have
According to (III) in Lemma 3, the first term and the second term on the right-hand side go to zero as The last term on the right-hand side also goes to zero as , according to (II) in Lemma 3.
By this theorem, immediately, we have
Corollary 1.
Assume that a sequence of events, , is exponentially independent under a Choquet expectation, , in the sense that
Then, Condition 9 in Theorem 1 holds.
4. Independence Cases
The notion of independence for random variables under non-additive expectation or non-additive probability is important. Motivated by mathematical finance and robust statistics, various different notions of independence have been investigated, for example, Peng’s independence, Marinacci pre-independence, and Puhalskii independence. It can be proven that all notions mentioned above satisfy Condition 9 in Theorem 1.
We now check that Condition 9 in Theorem 1 holds if event is Puhalskii-independent. The remaining cases can be verified in a similar manner. As defined below, Puhalskii independence implies the following:
Definition 5.
A sequence, , of events is said to be Puhalskii -independent if
This leads to the following lemma.
Lemma 4.
Let be a sequence of Puhalskii-independent events. Then, for any
Proof.
Let
Here, we have used Fubini’s theorem for Choquet expectation (see Ghirardato (1997) [] or Chateauneuf and Lefort (2008) []). □
Author Contributions
Writing—original draft, C.K.; Writing—review & editing, G.Z. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Data are contained within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
References
- Borel, E. Les probabilités dénombrables et leurs applications arith-métiq-ues. Rend. Circ. Mat. Palermo 1909, 27, 247–271. [Google Scholar] [CrossRef]
- Borel, E. Sur un probléme de probabilités relatif aux fractions continues. Math. Ann. 1912, 77, 578–587. [Google Scholar] [CrossRef]
- Cantelli, P. Sulla probabilitiá come limite della frequenza. Rend. Accad. Lincai Ser. 5 1917, 24, 39–45. [Google Scholar]
- Chandra, K. The Borel-Cantelli Lemma; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Barndorff-Nielsen, E. On the rate of growth of the partial maxima of a sequence of independent and identically distributed random variables. Math. Scand. 1961, 9, 383–394. [Google Scholar] [CrossRef]
- Martikainen, A.I.; Petrov, V.V. On the Borel–Cantelli lemma. Zap. Nauch. Semin. Leningr. Otd. Steklov Mat. Inst. 1990, 184, 200–207. (In Russian); English translation in: J. Math. Sci. 1994, 63, 540–544 [Google Scholar] [CrossRef]
- Balakrishnan, N.; Stepanov, A. A generalization of the Borel-Cantelli lemma. Math. Sci. 2010, 35, 61–62. [Google Scholar]
- Erdös, P.; Rényi, A. On Cantor’s series with convergent ∑1/qn. Ann. Univ. Sci. Budapest Eötvós. Sect. Math. 1959, 2, 93–109. [Google Scholar]
- Kochen, S.; Stone, C. A note on the Borel-Cantelli lemma. Ill. J. Math. 1964, 8, 248–251. [Google Scholar] [CrossRef]
- Petrov, V. A note on the Borel-Cantelli lemma. Stat. Probab. Lett. 2002, 58, 283–286. [Google Scholar] [CrossRef]
- Petrov, V. A generalization of the Borel-Cantelli lemma. Stat. Probab. Lett. 2004, 67, 233–239. [Google Scholar] [CrossRef]
- Yan, J. A simple proof of two generalized Borel-Cantelli lemmas. In Memorian Paul-Andre Meyer: Seminaire de Probabilitiés XXXIX; Lecture Notes in Mathematics No. 1874; Springer: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Xie, Y.Q. A bilateral inequality on nonnegative bounded random sequence. Stat. Probab. Lett. 2009, 79, 1577–1580. [Google Scholar] [CrossRef]
- Zong, G.; Chen, Z.; Lan, Y. Fubini-Like Theorem of Real-Valued Choquet Integrals for Set-Valued Mappings. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 2016, 24, 387–403. [Google Scholar] [CrossRef]
- Huber, P. The use of Choquet capacities in statistics. Bull. Int. Stat. Inst. 1973, 45, 181–191. [Google Scholar]
- Denneberg, D. Non-Additive Measure and Integral; Kluwer: Alphen aan den Rijn, The Netherlands, 1994. [Google Scholar]
- Wang, Z.; Klir, G. Generalized Measure Theory; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Peng, S. G-expectation, G-Brownian Motion and Related Stochastic Calculus of Ito type. In Proceedings of the 2005 Abel Symposium, Oslo, Norway, 29 July–4 August 2006. [Google Scholar]
- Peng, S. Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci. China Ser. A Math. 2009, 52, 1391–1411. [Google Scholar] [CrossRef]
- Peng, S. Nonlinear Expectations and Stochastic Calculus under Uncertainty: With Robust CLT and G-Brownian Motion; Springer: Berlin/Heidelberg, Germany, 2019. [Google Scholar]
- Torra, V.; Narukawa, Y.; Sugeno, M. Non-Additive Measures: Theory and Applications; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Billingsley, P. Probability and Measure, 3rd ed.; Wiley: New York, NY, USA, 1995. [Google Scholar]
- Maccheroni, F.; Marinacci, M. A strong law of large numbers for capacities. Ann. Probab. 2005, 33, 1171–1178. [Google Scholar] [CrossRef][Green Version]
- Puhalskii, A. Large Deviations and Idempotent Probability; Chapman and Hall/CRC: Boca Raton, FL, USA, 2001. [Google Scholar]
- Dhaene, J.; Denuit, M.; Goovaerts, M.J.; Kaas, R.; Vyncke, D. The concept of comonotonicity in actuarial science and finance: Theory. Insur. Math. Econ. 2002, 31, 3–33. [Google Scholar] [CrossRef]
- Ghirardato, P. On independence for non-additive measures with a Fubini theorem. J. Econom. Theory 1997, 73, 261–291. [Google Scholar] [CrossRef][Green Version]
- Chateauneuf, A.; Lefort, J. Some Fubini theorems on product σ-algebras for non-additive measures. Int. J. Approx. Reason. 2008, 48, 686–696. [Google Scholar] [CrossRef][Green Version]
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