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Conditional convergence for randomly weighted sums of random variables based on conditional residual hintegrability
Journal of Inequalities and Applications volume 2013, Article number: 122 (2013)
Abstract
Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ and $\{{A}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be two arrays of random variables defined on the same probability space $(\mathrm{\Omega},\mathcal{A},P)$ and ${\mathcal{B}}_{n}$ be subσalgebras of $\mathcal{A}$. Let $r>0$ be a constant. In this paper, we introduce some concepts of conditional residual hintegrability such as conditionally residually hintegrable relative to ${\mathcal{B}}_{n}$ concerning the array $\{{A}_{nk}\}$ with exponent r and conditionally strongly residually hintegrable relative to ${\mathcal{B}}_{n}$ concerning the array $\{{A}_{nk}\}$ 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 hintegrability with exponent r and conditional strongly residual hintegrability with exponent r, we obtain the conditional mean convergence and conditional almost sure convergence for randomly weighted sums.
MSC:60F15, 60F25.
1 Introduction
Throughout this paper, all the random variables are defined on the same probability space $(\mathrm{\Omega},\mathcal{A},P)$, and we let ℬ and ${\mathcal{B}}_{n}$ be subσalgebras of $\mathcal{A}$. Suppose that $\{{u}_{n},n\ge 1\}$ and $\{{v}_{n},n\ge 1\}$ are two sequences of integers (not necessary positive or finite) such that ${u}_{n}<{v}_{n}$ for all $n\ge 1$ and ${v}_{n}{u}_{n}\to \mathrm{\infty}$ as $n\to \mathrm{\infty}$. Let $\{{k}_{n},n\ge 1\}$ be a sequence of positive numbers such that ${k}_{n}\to \mathrm{\infty}$ as $n\to \mathrm{\infty}$. Let $I(A)$ be the indicator function of the set A and ${P}^{\mathcal{B}}(A)={E}^{\mathcal{B}}({I}_{A})$. Moreover, let $\{h(n),n\ge 1\}$ be an increasing sequence of positive constants with $h(n)\uparrow \mathrm{\infty}$ as $n\uparrow \mathrm{\infty}$.
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 GaltonWatson process (see, e.g., Basawa and Prakasa Rao [1]). Let $\{{Z}_{0}=1,{Z}_{n},n\ge 1\}$ be a GaltonWatson process with mean offspring Θ. This process can be studied by means of the following autoregressive type model:
where $\{{U}_{k},k\ge 0\}$ is the sequence of error random variables.
To estimate the mean offspring Θ from a realization $\{{Z}_{0}=1,{Z}_{n},n\ge 1\}$, the maximum likelihood estimator of Θ is ${\stackrel{\u02c6}{\mathrm{\Theta}}}_{n}={({\sum}_{k=1}^{n}{Z}_{k1})}^{1}({\sum}_{k=1}^{n}{Z}_{k})$, which coincides with the ’leastsquares’ estimator of Θ obtained by minimizing ${\sum}_{k=0}^{n}{U}_{k}^{2}$ with respect to Θ.
The study of asymptotic properties of ${\stackrel{\u02c6}{\mathrm{\Theta}}}_{n}$ leads to a conditional limit theorem since, as it is detailed in Basawa and Prakasa Rao [1], these asymptotic properties of ${\stackrel{\u02c6}{\mathrm{\Theta}}}_{n}$ depend on the event of nonextinction 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 hintegrability. 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 $(\mathrm{\Omega},\mathcal{A},P)$ with $E{X}^{2}<\mathrm{\infty}$ and $E{Y}^{2}<\mathrm{\infty}$. Prakasa Rao [6] defined the notion of conditional covariance of X and Y given ℬ (ℬcovariance for short) as
where ${E}^{\mathcal{B}}Z$ 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 ${Var}^{\mathcal{B}}X={Cov}^{\mathcal{B}}(X,X)$.
In the following, we present some concepts of uniform integrability.
The classical notion of uniform integrability of a sequence of integrable random variables $\{{X}_{n},n\ge 1\}$ 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 $\{{X}_{n},n\ge 1\}$ is said to be Cesàro uniformly integrable if
where $\{{m}_{n},n\ge 1\}$ is a sequence of positive integers such that ${m}_{n}\to \mathrm{\infty}$ as $n\to \mathrm{\infty}$.
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 $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of random variables, and let $\{{a}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of constants with ${\sum}_{k={u}_{n}}^{{v}_{n}}{a}_{nk}\le C$ for all $n\ge 1$ and some constant $C>0$. The array $\{{X}_{nk}\}$ is $\{{a}_{nk}\}$uniformly integrable if
Sung [10] gave a slight generalization and introduced the concept of Cesàrotype uniform integrability with exponent r as follows.
Definition 1.2 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of random variables and $r>0$. The array $\{{X}_{nk}\}$ is said to be Cesàrotype uniformly integral with exponent r if
and
Based on the Cesàrotype uniform integrability with exponent r for some $0<r<2$, Sung [10] obtained the weak law of large numbers for the array $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$.
Chandra and Goswami [11] introduced the concept of Cesàro αintegrability ($\alpha >0$) and showed that Cesàro αintegrability for any $\alpha >0$ is weaker than Cesàro uniform integrability. Under the condition of Cesàro αintegrability for some $\alpha >1/2$, they obtained the weak law of large numbers for a sequence of pairwise independent random variables. They also proved that Cesàro αintegrability 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 αintegrability, Chandra and Goswami [12] introduced the following concept of residual Cesàro $(\alpha ,r)$integrability, which is weaker than Cesàro αintegrability.
Definition 1.3 Let $\alpha \in (0,\mathrm{\infty})$, $r\in (0,\mathrm{\infty})$. A sequence of random variables $\{{X}_{n},n\ge 1\}$ is said to be residually Cesàro $(\alpha ,r)$integrable (RCI $(\alpha ,r)$ for short) if
and
It is easily seen that RCI $(\alpha ,r)$ integrability with $r=1$ is RCI $(\alpha )$ integrability (cf. Chandra and Goswami [12]).
Ordóñez Cabrera and Volodin [13] introduced the notion of hintegrability for an array of random variables concerning an array of constants $\{{a}_{nk}\}$ and showed that this concept is weaker than Cesàro uniform integrability, $\{{a}_{nk}\}$uniform integrability and Cesàro αintegrability. The notion of hintegrability for an array of random variables concerning an array of constants $\{{a}_{nk}\}$ is as follows.
Definition 1.4 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of random variables, and let $\{{a}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of constants with ${\sum}_{k={u}_{n}}^{{v}_{n}}{a}_{nk}\le C$ for all $n\ge 1$ and some constant $C>0$. The array $\{{X}_{nk}\}$ is said to be hintegrable concerning the array of constants $\{{a}_{nk}\}$ if
and
Sung et al. [14] generalized the notion of hintegrability and introduced the concept of hintegrability with exponent r. They also proved that the concept of hintegrability with exponent r is strictly weaker than the concept of Cesàrotype uniformly integral with exponent r. The concept of hintegrability with exponent r is as follows.
Definition 1.5 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of random variables and $r>0$. The array $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ is said to be hintegrable with exponent r if
and
At the same time, Yuan and Tao [15] introduced Rhintegrability concerning the array of constants $\{{a}_{nk}\}$ as follows.
Definition 1.6 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of random variables, and let $\{{a}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of constants. The array $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ is said to be residually hintegrable (Rhintegrable for short) concerning the array of constants $\{{a}_{nk}\}$ if
and
Inspired by the concepts above, recently, Ordóñez Cabrera et al. [4] introduced the notion of conditional residual hintegrability relative to the sequence $\{{\mathcal{B}}_{n}\}$ as follows.
Definition 1.7 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ and $\{{A}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be two arrays of random variables. The array $\{{X}_{nk}\}$ is said to be conditionally residually hintegrable relative to ${\mathcal{B}}_{n}$ (${\mathcal{B}}_{n}\text{}CR\text{}h$integrable for short) concerning the array $\{{A}_{nk}\}$ if
and
It is easily seen that if ${A}_{nk}\equiv {a}_{nk}$ are constants and ${\mathcal{B}}_{n}=\{\mathrm{\varnothing},\mathrm{\Omega}\}$ for all $n\ge 1$, then the concept of ${\mathcal{B}}_{n}\text{}CR\text{}h$integrability concerning the array $\{{A}_{nk}\}$ reduces to the concept of Rhintegrability concerning the array of constants $\{{a}_{nk}\}$.
To obtain a conditional strong convergence result, Ordóñez Cabrera et al. [4] introduced the following concept of conditional strongly residual hintegrability relative to the sequence ${\mathcal{B}}_{n}$, which is stronger than that of ${\mathcal{B}}_{n}\text{}CR\text{}h$integrability concerning the array $\{{A}_{nk}\}$.
Definition 1.8 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ and $\{{A}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be two arrays of random variables. The array $\{{X}_{nk}\}$ is said to be conditionally strongly residually hintegrable relative to ${\mathcal{B}}_{n}$ (${\mathcal{B}}_{n}\text{}CSR\text{}h$integrable for short) concerning the array $\{{A}_{nk}\}$ if
and
Inspired by the concepts above, we introduce some new concepts of conditional residual hintegrability such as conditionally residually hintegrable relative to ${\mathcal{B}}_{n}$ concerning the array $\{{A}_{nk}\}$ with exponent r and conditionally strongly residually hintegrable relative to ${\mathcal{B}}_{n}$ concerning the array $\{{A}_{nk}\}$ 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 hintegrability with exponent r and conditional strongly residual hintegrability 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 hintegrability relative to the sequence $\{{\mathcal{B}}_{n}\}$ with exponent r as follows.
Definition 2.1 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ and $\{{A}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be two arrays of random variables. Let $r>0$ be a constant. The array $\{{X}_{nk}\}$ is said to be conditionally residually hintegrable relative to ${\mathcal{B}}_{n}$ (${\mathcal{B}}_{n}\text{}CR\text{}h$integrable for short) concerning the array $\{{A}_{nk}\}$ 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 hintegrable relative to ${\mathcal{B}}_{n}$ (${\mathcal{B}}_{n}\text{}CR\text{}h$integrable for short) concerning the array $\{{A}_{nk}\}$, which was introduced by Ordóñez Cabrera et al. [4]. Just as Ordóñez Cabrera et al. [4] stated that the concept of ${\mathcal{B}}_{n}\text{}CR\text{}h$integrable concerning the array $\{{A}_{nk}\}$ is a conditional extension to the more general setting of randomly weighted sums of random variables of (i) the concept of residual Cesàro αintegrability introduced by Chandra and Goswami [12] and (ii) the concept of residual hintegrability concerning an array of constants introduced by Yuan and Tao [15], the concept of ${\mathcal{B}}_{n}\text{}CR\text{}h$integrable concerning the array $\{{A}_{nk}\}$ with exponent r is more general.
Remark 2.2 If ${A}_{nk}\equiv {a}_{nk}$ are constants, and ${\mathcal{B}}_{n}=\{\mathrm{\varnothing},\mathrm{\Omega}\}$ for all $n\ge 1$, then the preceding definition reduces to the following new concept of residual hintegrability concerning the array of constants $\{{a}_{nk}\}$ with exponent r.
Definition 2.2 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of random variables, and let $\{{a}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of constants. Let $r>0$ be a constant. The array $\{{X}_{nk}\}$ is said to be residually hintegrable (Rhintegrable for short) concerning the array $\{{a}_{nk}\}$ with exponent r if
and
Remark 2.3 Let $\{{h}_{1}(n),n\ge 1\}$ and $\{{h}_{2}(n),n\ge 1\}$ be two positive monotonically increasing to infinity sequences such that ${h}_{2}(n)\ge {h}_{1}(n)$ for all sufficiently large n. Then ${\mathcal{B}}_{n}\text{}CR\text{}{h}_{1}$integrability with exponent r implies ${\mathcal{B}}_{n}\text{}CR\text{}{h}_{2}$integrability with exponent r.
Remark 2.4 Take ${a}_{nk}=\frac{1}{{k}_{n}}$ for each k in Definition 2.2, where $\{{k}_{n}\}$ is a sequence of positive numbers such that ${k}_{n}\to \mathrm{\infty}$ as $n\to \mathrm{\infty}$. Note that
so the concept of Rhintegrable concerning the array $\{{a}_{nk}\}$ with exponent r is much weaker than the notion of hintegrability with exponent r, which was introduced by Sung [14].
Based on the condition of ${\mathcal{B}}_{n}\text{}CR\text{}h$integrability concerning the array $\{{A}_{nk}\}$ with exponent r, we study the conditional mean convergence for randomly weighted sums. Our main result is as follows.
Theorem 2.1 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of random variables, and let $\{{A}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of nonnegative random variables such that for each $n\ge 1$, $\{{A}_{nk},{u}_{n}\le k\le {v}_{n}\}$ are ${\mathcal{B}}_{n}$measurable. Suppose that

(i)
$$\{{X}_{nk}\}$$
is ${\mathcal{B}}_{n}\text{}CR\text{}h$integrable concerning the array $\{{A}_{nk}\}$ with exponent $0<r<1$;

(ii)
$$h(n){sup}_{{u}_{n}\le k\le {v}_{n}}{A}_{nk}\to 0$$
a.s. as $n\to \mathrm{\infty}$.
Let ${S}_{n}={\sum}_{k={u}_{n}}^{{v}_{n}}{A}_{nk}^{1/r}{X}_{nk}$, $n\ge 1$. Then ${E}^{{\mathcal{B}}_{n}}{{S}_{n}}^{r}\to 0$ a.s. as $n\to \mathrm{\infty}$.
Proof For each $n\ge 1$ and ${u}_{n}\le k\le {v}_{n}$, we define, by using the method of continuous truncation, the following:
and ${Z}_{nk}={X}_{nk}{Y}_{nk}$. Hence, we can write that
and we estimate the conditional expectation of each of these terms separately. With regard to ${S}_{2n}$, since ${Z}_{nk}=({X}_{nk}{h}^{1/r}(n))I({{X}_{nk}}^{r}>h(n))$, we have by ${C}_{r}$’s inequality and Definition 1.8 that
For ${S}_{1n}$, we initially prove ${E}^{{\mathcal{B}}_{n}}{S}_{1n}\to 0$ a.s. as $n\to \mathrm{\infty}$. 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 ${C}_{r}$’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 hintegrability relative to the sequence ${\mathcal{B}}_{n}$ with exponent r as follows.
Definition 3.1 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ and $\{{A}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be two arrays of random variables. Let $r>0$ be a constant. The array $\{{X}_{nk}\}$ is said to be conditionally strongly residually hintegrable relative to ${\mathcal{B}}_{n}$ (${\mathcal{B}}_{n}\text{}CSR\text{}h$integrable for short) concerning the array $\{{A}_{nk}\}$ 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 hintegrable relative to ${\mathcal{B}}_{n}$ (${\mathcal{B}}_{n}\text{}CSR\text{}h$integrable, for short) concerning the array $\{{A}_{nk}\}$, which was introduced by Ordóñez Cabrera et al. [4].
Remark 3.2 If ${A}_{nk}\equiv {a}_{nk}$ are constants, and ${\mathcal{B}}_{n}=\{\mathrm{\varnothing},\mathrm{\Omega}\}$ for all $n\ge 1$, then the preceding definition reduces to the following new concept of strongly residual hintegrability concerning the array of constants $\{{a}_{nk}\}$ with exponent r.
Definition 3.2 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of random variables, and let $\{{a}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of constants. Let $r>0$ be a constant. The array $\{{X}_{nk}\}$ is said to be strongly residually hintegrable (SRhintegrable for short) concerning the array $\{{a}_{nk}\}$ with exponent r if
and
Remark 3.3 It is easily seen that the concept of ${\mathcal{B}}_{n}\text{}CSR\text{}h$integrability with exponent r is stronger than the concept of ${\mathcal{B}}_{n}\text{}CR\text{}h$integrability with exponent r. Likewise, the unconditional concept of SRhintegrability with exponent r is stronger than the concept of Rhintegrability with exponent r.
We now establish a strong version of Theorem 2.1 under the condition of $\mathcal{B}\text{}CSR\text{}h$integrability (i.e., when ${\mathcal{B}}_{n}=\mathcal{B}$, a subσalgebra of $\mathcal{A}$, for all $n\ge 1$).
Theorem 3.1 Let $\{{X}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of random variables, and let $\{{A}_{nk},{u}_{n}\le k\le {v}_{n},n\ge 1\}$ be an array of nonnegative ℬmeasurable random variables. Suppose that

(i)
$$\{{X}_{nk}\}$$
is $\mathcal{B}\text{}CSR\text{}h$integrable concerning the array $\{{A}_{nk}\}$ with exponent $0<r<1$;

(ii)
$${\sum}_{n=1}^{\mathrm{\infty}}{(h(n){sup}_{{u}_{n}\le k\le {v}_{n}}{A}_{nk})}^{\frac{1r}{r}}<\mathrm{\infty}$$
a.s.
Then ${S}_{n}={\sum}_{k={u}_{n}}^{{v}_{n}}{A}_{nk}^{1/r}{X}_{nk}\to 0$ a.s. as $n\to \mathrm{\infty}$.
Proof We use the same notations as those in Theorem 2.1 and set ${\mathcal{B}}_{n}=\mathcal{B}$ for each $n\ge 1$. Then ${S}_{n}={S}_{1n}+{S}_{2n}$ for each $n\ge 1$, and we estimate each of these terms separately.
Condition (i) implies via the nonnegativity of every summand that
which implies that
Note that ${Z}_{nk}=({X}_{nk}{h}^{1/r}(n))I({{X}_{nk}}^{r}>h(n))$, we have by (3.1) that
which implies that ${S}_{2n}\to 0$ a.s. as $n\to \mathrm{\infty}$.
Now we prove that ${S}_{1n}\to 0$ a.s. as $n\to \mathrm{\infty}$. By the conditional Markov’s inequality, we have for all $\epsilon >0$ 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 BorelCantelli lemma that
which implies that ${S}_{1n}\to 0$ a.s. as $n\to \mathrm{\infty}$ since the ${P}^{\mathcal{B}}$null sets and the Pnull sets coincide (see, e.g., Yuan and Xie, [16]).
Therefore, ${S}_{n}={S}_{1n}+{S}_{2n}\to 0$ a.s. as $n\to \mathrm{\infty}$. 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 hintegrability. J Inequal Appl 2013, 122 (2013). https://doi.org/10.1186/1029242X2013122
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Keywords
 uniform integrability
 randomly weighted sums
 conditional mean convergence
 conditional almost sure convergence
 conditionally residually integrable
 conditionally strongly residually integrable