Abstract
Let and be two arrays of random variables defined on the same probability space and be sub-σ-algebras of . Let be a constant. In this paper, we introduce some concepts of conditional residual h-integrability such as conditionally residually h-integrable relative to concerning the array with exponent r and conditionally strongly residually h-integrable relative to concerning the array with exponent r. These concepts are more general than some known setting of randomly weighted sums of random variables. Based on the conditions of conditional residual h-integrability with exponent r and conditional strongly residual h-integrability with exponent r, we obtain the conditional mean convergence and conditional almost sure convergence for randomly weighted sums.
MSC:60F15, 60F25.
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1 Introduction
Throughout this paper, all the random variables are defined on the same probability space , and we let ℬ and be sub-σ-algebras of . Suppose that and are two sequences of integers (not necessary positive or finite) such that for all and as . Let be a sequence of positive numbers such that as . Let be the indicator function of the set A and . Moreover, let be an increasing sequence of positive constants with as .
1.1 Brief review
Conditional limit theorems play a key role in the study of statistical inference. A typical example of statistical application of conditional limit theorems is in the study of statistical inference for some branching processes such as the Galton-Watson process (see, e.g., Basawa and Prakasa Rao [1]). Let be a Galton-Watson process with mean offspring Θ. This process can be studied by means of the following autoregressive type model:
where is the sequence of error random variables.
To estimate the mean offspring Θ from a realization , the maximum likelihood estimator of Θ is , which coincides with the ’least-squares’ estimator of Θ obtained by minimizing with respect to Θ.
The study of asymptotic properties of leads to a conditional limit theorem since, as it is detailed in Basawa and Prakasa Rao [1], these asymptotic properties of depend on the event of non-extinction of the process. For more details about the conditional limit theorem, one can refer to Roussas [2], Leek [3], Ordóñez Cabrera et al. [4], and so forth.
The main purpose of the paper is to introduce some new concepts of conditional residual h-integrability. Based on these conditions, we study the conditional mean convergence and conditional almost sure convergence for randomly weighted sums of random variables.
1.2 Some concepts of integrability
It is well known that the notion of uniform integrability plays the central role in establishing weak laws of large numbers. In this section, we recall some concepts of integrability.
Firstly, we recall the notion of conditional covariance. The interested reader can find further results in Chow and Teicher [5] and Roussas [2]. Let X and Y be random variables defined on a probability space with and . Prakasa Rao [6] defined the notion of conditional covariance of X and Y given ℬ (ℬ-covariance for short) as
where denotes the conditional expectation of a random variable Z given ℬ. In contrast to the ordinary concept of variance, conditional variance of X given ℬ is defined as .
In the following, we present some concepts of uniform integrability.
The classical notion of uniform integrability of a sequence of integrable random variables is defined through the condition
Landers and Rogge [7] proved that the uniform integrability condition is sufficient in order that a sequence of pairwise independent random variables verifies the weak laws of large numbers.
Chandra [8] introduced the notion of Cesàro uniform integrability which is weaker than uniform integrability. A sequence of integrable random variables is said to be Cesàro uniformly integrable if
where is a sequence of positive integers such that as .
Ordóñez Cabrera [9], by studying the weak convergence for weighted sums of random variables, introduced the condition of uniform integrability concerning the weights, which is weaker than uniform integrability and includes the Cesàro uniform integrability as a special case.
Definition 1.1 Let be an array of random variables, and let be an array of constants with for all and some constant . The array is -uniformly integrable if
Sung [10] gave a slight generalization and introduced the concept of Cesàro-type uniform integrability with exponent r as follows.
Definition 1.2 Let be an array of random variables and . The array is said to be Cesàro-type uniformly integral with exponent r if
and
Based on the Cesàro-type uniform integrability with exponent r for some , Sung [10] obtained the weak law of large numbers for the array .
Chandra and Goswami [11] introduced the concept of Cesàro α-integrability () and showed that Cesàro α-integrability for any is weaker than Cesàro uniform integrability. Under the condition of Cesàro α-integrability for some , they obtained the weak law of large numbers for a sequence of pairwise independent random variables. They also proved that Cesàro α-integrability for appropriate α is also sufficient for the weak law of large numbers to hold for certain special dependent sequences of random variables.
Inspired by the concept of Cesàro α-integrability, Chandra and Goswami [12] introduced the following concept of residual Cesàro -integrability, which is weaker than Cesàro α-integrability.
Definition 1.3 Let , . A sequence of random variables is said to be residually Cesàro -integrable (RCI for short) if
and
It is easily seen that RCI integrability with is RCI integrability (cf. Chandra and Goswami [12]).
Ordóñez Cabrera and Volodin [13] introduced the notion of h-integrability for an array of random variables concerning an array of constants and showed that this concept is weaker than Cesàro uniform integrability, -uniform integrability and Cesàro α-integrability. The notion of h-integrability for an array of random variables concerning an array of constants is as follows.
Definition 1.4 Let be an array of random variables, and let be an array of constants with for all and some constant . The array is said to be h-integrable concerning the array of constants if
and
Sung et al. [14] generalized the notion of h-integrability and introduced the concept of h-integrability with exponent r. They also proved that the concept of h-integrability with exponent r is strictly weaker than the concept of Cesàro-type uniformly integral with exponent r. The concept of h-integrability with exponent r is as follows.
Definition 1.5 Let be an array of random variables and . The array is said to be h-integrable with exponent r if
and
At the same time, Yuan and Tao [15] introduced R-h-integrability concerning the array of constants as follows.
Definition 1.6 Let be an array of random variables, and let be an array of constants. The array is said to be residually h-integrable (R-h-integrable for short) concerning the array of constants if
and
Inspired by the concepts above, recently, Ordóñez Cabrera et al. [4] introduced the notion of conditional residual h-integrability relative to the sequence as follows.
Definition 1.7 Let and be two arrays of random variables. The array is said to be conditionally residually h-integrable relative to (-integrable for short) concerning the array if
and
It is easily seen that if are constants and for all , then the concept of -integrability concerning the array reduces to the concept of R-h-integrability concerning the array of constants .
To obtain a conditional strong convergence result, Ordóñez Cabrera et al. [4] introduced the following concept of conditional strongly residual h-integrability relative to the sequence , which is stronger than that of -integrability concerning the array .
Definition 1.8 Let and be two arrays of random variables. The array is said to be conditionally strongly residually h-integrable relative to (-integrable for short) concerning the array if
and
Inspired by the concepts above, we introduce some new concepts of conditional residual h-integrability such as conditionally residually h-integrable relative to concerning the array with exponent r and conditionally strongly residually h-integrable relative to concerning the array with exponent r. These concepts are more general than some known setting of randomly weighted sums of random variables. Based on the conditions of conditional residual h-integrability with exponent r and conditional strongly residual h-integrability with exponent r, we study the conditional mean convergence and conditional almost sure convergence for randomly weighted sums.
2 Conditional mean convergence for randomly weighted sums
In this section, we study the conditional mean convergence for randomly weighted sums. Inspired by the concepts stated in Section 1, we introduce the notion of conditional residual h-integrability relative to the sequence with exponent r as follows.
Definition 2.1 Let and be two arrays of random variables. Let be a constant. The array is said to be conditionally residually h-integrable relative to (-integrable for short) concerning the array with exponent r if
and
Remark 2.1 In Definition 2.1, if the exponent r takes value 1, then we can get the concept of conditionally residually h-integrable relative to (-integrable for short) concerning the array , which was introduced by Ordóñez Cabrera et al. [4]. Just as Ordóñez Cabrera et al. [4] stated that the concept of -integrable concerning the array is a conditional extension to the more general setting of randomly weighted sums of random variables of (i) the concept of residual Cesàro α-integrability introduced by Chandra and Goswami [12] and (ii) the concept of residual h-integrability concerning an array of constants introduced by Yuan and Tao [15], the concept of -integrable concerning the array with exponent r is more general.
Remark 2.2 If are constants, and for all , then the preceding definition reduces to the following new concept of residual h-integrability concerning the array of constants with exponent r.
Definition 2.2 Let be an array of random variables, and let be an array of constants. Let be a constant. The array is said to be residually h-integrable (R-h-integrable for short) concerning the array with exponent r if
and
Remark 2.3 Let and be two positive monotonically increasing to infinity sequences such that for all sufficiently large n. Then -integrability with exponent r implies -integrability with exponent r.
Remark 2.4 Take for each k in Definition 2.2, where is a sequence of positive numbers such that as . Note that
so the concept of R-h-integrable concerning the array with exponent r is much weaker than the notion of h-integrability with exponent r, which was introduced by Sung [14].
Based on the condition of -integrability concerning the array with exponent r, we study the conditional mean convergence for randomly weighted sums. Our main result is as follows.
Theorem 2.1 Let be an array of random variables, and let be an array of nonnegative random variables such that for each , are -measurable. Suppose that
-
(i)
is -integrable concerning the array with exponent ;
-
(ii)
a.s. as .
Let , . Then a.s. as .
Proof For each and , we define, by using the method of continuous truncation, the following:
and . Hence, we can write that
and we estimate the conditional expectation of each of these terms separately. With regard to , since , we have by ’s inequality and Definition 1.8 that
For , we initially prove a.s. as . Noting that
we have by (2.1), condition (ii) and Definition 1.8 that
which implies that
By Jensen’s inequality for conditional expectations (see, e.g., Chow and Teicher [5], p. 217), we have by (2.2) that
Therefore, we have by ’s inequality that
This completes the proof of the theorem. □
3 Conditional almost sure convergence for randomly weighted sums
To obtain a conditional strong convergence result, we introduce the concept of conditional strongly residual h-integrability relative to the sequence with exponent r as follows.
Definition 3.1 Let and be two arrays of random variables. Let be a constant. The array is said to be conditionally strongly residually h-integrable relative to (-integrable for short) concerning the array with exponent r if
and
Remark 3.1 In Definition 3.1, if the exponent r takes value 1, then we can get the conditionally strongly residually h-integrable relative to (-integrable, for short) concerning the array , which was introduced by Ordóñez Cabrera et al. [4].
Remark 3.2 If are constants, and for all , then the preceding definition reduces to the following new concept of strongly residual h-integrability concerning the array of constants with exponent r.
Definition 3.2 Let be an array of random variables, and let be an array of constants. Let be a constant. The array is said to be strongly residually h-integrable (SR-h-integrable for short) concerning the array with exponent r if
and
Remark 3.3 It is easily seen that the concept of -integrability with exponent r is stronger than the concept of -integrability with exponent r. Likewise, the unconditional concept of SR-h-integrability with exponent r is stronger than the concept of R-h-integrability with exponent r.
We now establish a strong version of Theorem 2.1 under the condition of -integrability (i.e., when , a sub-σ-algebra of , for all ).
Theorem 3.1 Let be an array of random variables, and let be an array of nonnegative ℬ-measurable random variables. Suppose that
-
(i)
is -integrable concerning the array with exponent ;
-
(ii)
a.s.
Then a.s. as .
Proof We use the same notations as those in Theorem 2.1 and set for each . Then for each , and we estimate each of these terms separately.
Condition (i) implies via the nonnegativity of every summand that
which implies that
Note that , we have by (3.1) that
which implies that a.s. as .
Now we prove that a.s. as . By the conditional Markov’s inequality, we have for all that
where the third inequality follows from (2.1) and the last inequality follows from condition (ii) and Definition 3.1. Hence, we have by (3.2) and the conditional Borel-Cantelli lemma that
which implies that a.s. as since the -null sets and the P-null sets coincide (see, e.g., Yuan and Xie, [16]).
Therefore, a.s. as . This completes the proof of the theorem. □
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Acknowledgements
The authors are most grateful to the editor and anonymous referee for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11201001, 11171001, 11126176, 11226207), the Natural Science Foundation of Anhui Province (1308085QA03, 11040606M12, 1208085QA03), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20093401120001), the 211 project of Anhui University, the Youth Science Research Fund of Anhui University and the Students Science Research Training Program of Anhui University (KYXL2012007).
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Shen, A., Wu, R., Chen, Y. et al. Conditional convergence for randomly weighted sums of random variables based on conditional residual h-integrability. J Inequal Appl 2013, 122 (2013). https://doi.org/10.1186/1029-242X-2013-122
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DOI: https://doi.org/10.1186/1029-242X-2013-122