Equivalent Conditions of Complete <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 10.3455 13.4804" width="10.3455pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg>th Moment Convergence for Weighted Sums of I. I. D. Random Variables under Sublinear Expectations

Xu, Mingzhou; Cheng, Kun

doi:https://doi.org/10.1155/2021/7471550

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 7471550 | https://doi.org/10.1155/2021/7471550

Equivalent Conditions of Complete th Moment Convergence for Weighted Sums of I. I. D. Random Variables under Sublinear Expectations

Mingzhou Xu¹and Kun Cheng¹

Academic Editor: Ya Jia

Received20 Jul 2021

Revised19 Sept 2021

Accepted30 Sept 2021

Published18 Oct 2021

Abstract

We investigate the complete th moment convergence for weighted sums of independent, identically distributed random variables under sublinear expectations space. Using moment inequality and truncation methods, we prove the equivalent conditions of complete th moment convergence of weighted sums of independent, identically distributed random variables under sublinear expectations space, which complement the corresponding results obtained in Guo and Shan (2020).

1. Introduction

Peng [1, 2] presented the concept of the sublinear expectation space to study the uncertainty of probability and distribution. The seminal work of Peng [1, 2] encourages people to study limit theorems under sublinear expectations space. Zhang [3–5] proved results including exponential inequalities, Rosenthal’s inequalities, and Donsker’s invariance principle under sublinear expectations. Wu [6] obtained precise asymptotics for complete integral convergence. Xu and Cheng [7] studied precise asymptotics in the law of iterated logarithm under sublinear expectations. The interested reader could refer to Xu and Zhang [8, 9], Chen [10], Gao and Xu [11], Fang et al. [12], Hu et al. [13], Hu and Yang [14], Huang and Wu [15], Kuczmaszewska [16], Ma and Wu [17], Wang and Wu [18], Wu and Jiang [19], Yu and Wu [20], Zhang [21], Zhong and Wu [22], and references therein for more limit theorems under sublinear expectations.

Recently, Guo and Shan [23] studied equivalent conditions of complete th moment convergence for weighted sums of sequences of negatively orthant dependent random variables. Xu and Cheng [24] obtained equivalent conditions of complete convergence for weighted sums of sequences of i. i. d. random variables under sublinear expectations. Motivated by the work of Guo and Shan [23] and Xu and Cheng [24], here we try to prove the equivalent conditions of complete th moment convergence of weighted sums of independent, identically distributed random variables under sublinear expectations space, which complement the corresponding results obtained in Guo and Shan [23], also extend results in Xu and Cheng [24] from complete convergence to complete th moment convergence.

We organized the rest of this paper as follows. In Section 2, we give necessary basic notions, concepts, and relevant properties and present necessary lemmas under sublinear expectations. In Section 3, we give our main results, Theorems 1–4, whose proofs are presented in Section 4.

2. Preliminaries

As in Xu and Cheng [24], we adopt similar notations as in the work by Peng [2] and Chen [10]. Suppose that is a given measurable space. We assume that is a subset of all random variables on such that (cf. [10]), where , and implies , for each , where denotes the linear space of (local Lipschitz) function satisfyingfor some and depending on .

Definition 1. A sublinear expectation on is a functional satisfying the following properties: for all , we have(a)Monotonicity: if , then (b)Constant preserving: , (c)Positive homogeneity: , (d)Subadditivity: whenever is not of the form or

A set function is said to be a capacity if it obeys(a) and .(b), and . Moreover, if is continuous, then should satisfy(c) if .(d) if .

A capacity is called subadditive if , .

In this paper, given a sublinear expectation space , set , (see (8) and the definitions of above (8) in [4]). Clearly, is a subadditive capacity. Denote the Choquet expectations by

Suppose that , , and , , are two random vectors on . is called to be independent of if, for each Borel-measurable function on with ; for each , we have whenever for each and (see Definition 2.5 in [10]). is called a sequence of independent random variables if is independent of , for each .

Assume that and are two -dimensional random vectors defined, respectively, in sublinear expectation spaces and . They are called identically distributed if, for every Borel-measurable function such that ,whenever the sublinear expectations are finite. is called to be identically distributed if, for each , and are identically distributed.

In the sequel, we suppose that is countably subadditive, i.e., , whenever , , and , , . Let denote a positive constant which may differ from line to line. or represents the indicator function of , means that there exists a constant such that for large sufficiently, and means that and . We use for .

We first present several necessary lemmas to prove our main results. By using Corollary 2.2, Theorem 2.3 in [5], the proofs of Theorem 3 in [25], and Minkowski’s inequality under sublinear expectations, we see that the following lemma holds.

Lemma 1 (cf. Lemma 2.3 in [24]). Suppose that is a sequence of independent random variables under sublinear expectation space with , , , and . Then,where depends on only.

Proof. For readers’ convenience, we give complete proofs here. First, for and , set and , andwhere depends on determined as in (2.8) of Corollary 2.2 (b) in [5]. By Corollary 2.2 in [5], we know that, for all and ,Obviously, for and ,Set , and for , , where is the integer part of . implies that . Now, it is enough to prove that, for and ,As in the proof of Theorem 4 of Móricz [25], let be given. Obviously, or . For and , we see thatwhence, for such ’s,Since, for , we have , hence, for ,Thus,and by Minkowski’s inequality under sublinear expectations (see (4.10) in Proposition 4.2 of Chapter I of [2]),Suppose now that (8) holds for . Then, by the choice of , we see thatBy these two inequalities above and (7), we see thatFinally, (6) impliesBy (13)–(16), we conclude thatwhich implies the result. Hence, by (6), the conclusion of (8) is true for . By induction, (8) holds for all . The proof is complete.

Lemma 2 (see Lemma 2.4 in [24, 26]). Let be a sequence of independent random variables under sublinear expectation space . Then, for all and ,

Remark 1. In the proofs of Lemma 2.4 in [24, 26], by independence of , for constant and (Definition 2.5 in [10]), we see thatwhich implies that Lemma 2 is valid. The difference between the sublinear expectations and linear expectations could be implied by the subtle observation thatwhich is not necessarily equal to .

Lemma 3. Let be a random variable under sublinear expectation space , , , and . Then, the following are equivalent:(i)(ii)

Proof. The proof is finished.

Lemma 4. Let be a random variable under sublinear expectation space , , , and . Then, the following is equivalent:(i)(ii)

Proof. This finishes the proof.

3. Main Results

We state our main results, the proofs of which will be given in Section 4.

Theorem 1. Let be a sequence of independent random variables, identically distributed as under sublinear expectation space . Assume that , , and , and suppose that for . Suppose that is a triangular array of real numbers. Then, the following is equivalent:(i)(ii)

Remark 2. If is classic probability space, then Theorem 1 recovers Theorem 7 in [23] in case in which is a sequence of independent random variables, identically distributed as . As pointed in Hossein and Nezakati [27], why we need th moment convergence under sublinear expectations, the complete moment covergence is a more general expression than complete convergence in theory and practice, and the interested reader also could refer to the first study in complete moment convergence by Chow [28].

Remark 3. Under the same conditions in Theorem 1, if (28) holds, by the same proof of Remark 3.2 of Hossein and Nezakati [27], we see that, for all ,Moreover, if is continuous, by (29), we see thatIn (30), if , , we conclude that , which is similar to Theorem 1 of Zhang and Lin [29].

Theorem 2. Let be a sequence of independent random variables, identically distributed as under sublinear expectation space . Assume that , , and , and suppose that for . Suppose that is a triangular array of real numbers. Then, (28) is equivalent to

Theorem 3. Let be a sequence of independent random variables, identically distributed as under sublinear expectation space . Assume that , , and , and suppose that for . Suppose that is a triangular array of real numbers. Then, (28) is equivalent to

As in the proofs of Theorems 1–3, we can get the following corollary.

Corollary 1. Let be a sequence of independent random variables, identically distributed as under sublinear expectation space . Assume that , , and , and suppose that , for . Suppose that is a triangular array of real numbers. Then,(i)(27) is equivalent to(ii)(31) is equivalent to (33) when .(iii)(32) is equivalent to (33) when .

The following theorem is complete th moment convergence on Cesàro summation of independent, identically distributed random variables under.

Theorem 4. Let be a sequence of independent random variables, identically distributed as under sublinear expectation space . Assume that , , , and , and suppose that , for . Suppose , and . Then,(i)(27) is equivalent to when .(ii)(31) is equivalent to (34) when .(iii)(34) is equivalent towhen .

Remark 4. If is classic probability space, then Theorems 1–3 and Corollary 1 recover, respectively, Theorems 10, 11, and 14 and Corollary 13 in Guo and Shan [23] in case in which is a sequence of independent random variables, identically distributed as .

4. Proofs of the Main Results

Proof. of Theorem 1. We first prove that (27) implies (28). Notice thatFrom Xu and Cheng [24] and the fact that (27) implies , we see that . We next establish . Choose , and is small sufficiently and integer is large enough. For every , , we note the fact that is large sufficiently to guarantee . Since the first finite terms of the series do not affect the convergence of the series, without loss of restictions, we could assume the definitions of below are meaningful. WriteObserve that . Notice thatConsequently, to prove (28), we only need to prove thatFrom the definition of , we deduce thatObserve that implies . Therefore,From , we see thatHence, from Lemma 3 and (27), we obtain . From the definition of , we conclude that . From the subadditivity of capacity and Definition 2.5 in [10], it follows thatFrom , we obtain . Since (27) implies , by Markov’s inequality under sublinear expectations (cf. (9) in Hu and Wu [30]) and (43), we conclude thatNotice that and ; we could choose small sufficiently and integer large enough such that and . Hence, from (44), we obtain . Similarly, we can get . In order to estimate , we first prove thatObserve that (27), Lemma 4.5 in Zhang [4], and Hölder’s inequality under sublinear expectations imply that and . When , observe that and , by Hölder’s inequality, we see thatObserving that , by (46), for any , we obtainWhen , noticing that and by choosing small sufficiently such that , we see thatObserving that , we haveThus, to establish , we only need to prove thatObserve that is independent, identically distributed under . By Markov’s inequality under sublinear expectations, ’s inequality, and Lemma 1, we conclude that, for a suitably large , which will be determined later:Choosing large sufficiently such that and , we haveWhen , (27) implies that . Observing that , we could choose large enough such that , . Then, by ,When , choosing large enough such that and , we haveConsequently, by (51)–(54), we get . Now, we prove that (28) implies (27). By Markov’s inequality under sublinear expectations (see (8) of Wu [6]), (28), and Lemma 4.5 (iii) in [4],Sinceby (55) and similar proofs of (3.17) in [31], we conclude thatTherefore, by (57) and Lemma 2, we obtainNow, combining (58) with (28) givesFrom the proof of (42), we conclude that (59) is equivalent to (27).

Proof. of Theorem 2. As in the proof of Theorem 1, with Lemma 4 in place of Lemma 3, and usingwhen , we could prove Theorem 2. Hence, the proof is omitted.

Proof. of Theorem 3. As in the proof of Theorem 1, usingand within place of (41), we could prove Theorem 3. Thus, the proof is omitted.

Proof. of Theorem 4. Setting , and , we observe that , , , and . As in the proof of Theorem 14 in [23], letting in Corollary 1, we finish the proof of Theorem 4.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and read and approved the final manuscript.

Acknowledgments

This research was supported by Doctoral Scientific Research Starting Foundation of Jingdezhen Ceramic University (no. 102/01003002031), Scientific Program of Department of Education of Jiangxi Province of China (nos. GJJ190732 and GJJ180737), Natural Science Foundation Program of Jiangxi Province (no. 20202BABL211005), and National Natural Science Foundation of China (no. 61662037).

References

S. Peng, “G-expectation, G-brownian motion and related stochastic calculus of itô type,” Stochastic Analysis and Applications, vol. 2, no. 4, pp. 541–567, 2007.
View at: Publisher Site | Google Scholar
S. G. Peng, “Nonlinear expectations and stochastic calculus under uncertainty,” 2010, https://arxiv.org/abs/1002.4546.
View at: Google Scholar
L.-X. Zhang, “Donsker’s invariance principle under the sub-linear expectation with an application to chung;s law of the iterated logarithm,” Communications in Mathematics and Statistics, vol. 3, no. 2, pp. 187–214, 2015.
View at: Publisher Site | Google Scholar
L. Zhang, “Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm,” Science China Mathematics, vol. 59, no. 12, pp. 2503–2526, 2016a.
View at: Publisher Site | Google Scholar
L. X. Zhang, “Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications,” Science China Mathematics, vol. 59, no. 4, pp. 759–768, 2016b.
View at: Publisher Site | Google Scholar
Q. Y. Wu, “Precise asymptotics for complete integral convergence under sublinear expectations,” Mathematical Problems in Engineering, vol. 2020, Article ID 3145935, 13 pages, 2020.
View at: Publisher Site | Google Scholar
M. Z. Xu and K. Cheng, “Precise asymptotics in the law of the iterated logarithm under sublinear expectations,” Mathematical Problems in Engineering, vol. 2021, Article ID 6691857, 9 pages, 2021.
View at: Publisher Site | Google Scholar
J. P. Xu and L. X. Zhang, “Three series theorem for independent random variables under sub-linear expectations with applications,” Acta Mathematica Sinica, English Series, vol. 35, no. 2, pp. 172–184, 2019.
View at: Publisher Site | Google Scholar
J.-P. Xu and L.-X. Zhang, “The law of logarithm for arrays of random variables under sub-linear expectations,” Acta Mathematicae Applicatae Sinica, English Series, vol. 36, no. 3, pp. 670–688, 2020.
View at: Publisher Site | Google Scholar
Z. Chen, “Strong laws of large numbers for sub-linear expectations,” Science China Mathematics, vol. 59, no. 5, pp. 945–954, 2016.
View at: Publisher Site | Google Scholar
F. Q. Gao and M. Z. Xu, “Large deviations and moderate deviations for independent random variables under sublinear expectations (in Chinese),” Science China Mathematics, vol. 41, no. 4, pp. 337–352, 2011.
View at: Google Scholar
X. Fang, S. G. Peng, Q. M. Shao, and Y. S. Song, “Limit theorems with rate of convergence under sublinear expectations,” Bernoulli, vol. 25, no. 4A, pp. 2564–2596, 2018.
View at: Google Scholar
F. Hu, Z. J. Chen, and D. F. Zhang, “How big are the increments of G-Brownian motion,” Science China Mathematics, vol. 57, no. 8, pp. 1686–1700, 2014.
View at: Publisher Site | Google Scholar
Z.-C. Hu and Y.-Z. Yang, “Some inequalities and limit theorems under sublinear expectations,” Acta Mathematicae Applicatae Sinica, English Series, vol. 33, no. 2, pp. 451–462, 2017.
View at: Publisher Site | Google Scholar
W. H. Huang and P. Y. Wu, “Strong laws of large numbers for general random variables in sublinear expectation spaces,” Journal of Inequalities and Applications, vol. 2019, Article ID 143, 18 pages, 2019.
View at: Publisher Site | Google Scholar
A. Kuczmaszewska, “Complete convergence for widely acceptable random variables under the sublinear expecations,” Journal of Mathematical Analysis and Applications, vol. 484, no. 1, Article ID 123662, p. 15, 2020.
View at: Publisher Site | Google Scholar
X. C. Ma and Q. Y. Wu, “Limiting behavior of weighted sums of extended negatively dependent random variables under sublinear expectations,” Advances in Mathematics, vol. 48, no. 2, pp. 89–96, 2018.
View at: Google Scholar
W. J. Wang and Q. Y. Wu, “Almost sure convergence of weighted sums for extended negatively dependent random variables under sub-linear expectations,” Mathematica Applicata, vol. 32, no. 2, pp. 382–391, 2019.
View at: Google Scholar
Q. Wu and Y. Jiang, “Strong law of large numbers and Chover’s law of the iterated logarithm under sub-linear expectations,” Journal of Mathematical Analysis and Applications, vol. 460, no. 1, pp. 252–270, 2018.
View at: Publisher Site | Google Scholar
D. L. Yu and Q. Y. Wu, “Marcinkiewicz type complete convergence for weighted sums under sub-linear expectations,” Journal of University of Science and Technology of China, vol. 48, no. 2, pp. 89–96, 2018.
View at: Google Scholar
L. X. Zhang, “Strong limit theorems for extended independent and extended negatively random variables under non-linear expectations,” 2016, https://arxiv.org/abs/1608.0071v1.
View at: Google Scholar
H. Y. Zhong and Q. Y. Wu, “Complete convergence and complete moment convergence for weighted sums of extended negatively dependent random variables under sub-linear expectation,” Journal of Inequalities and Applications, vol. 2017, Article ID 261, 14 pages, 2017.
View at: Publisher Site | Google Scholar
M. L. Guo and S. W. Shan, “Equivalent conditions of complete qth monent convergence for weighted sums of sequences of negatively orthant dependent raodom variables,” Chinese Journal of Applied Probabability and Statistics, vol. 36, no. 4, pp. 381–392, 2020.
View at: Google Scholar
M. Z. Xu and K. Cheng, “Equivalent conditions of complete convergence for weighted sums of sequences of i. i. d. random variables under sublinear expectations,” 2021, https://arxiv.org/abs/2108.12085v1.
View at: Google Scholar
F. Móricz, “Moment inequalities and the strong law of large numbers,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 35, no. 4, pp. 299–314, 1976.
View at: Google Scholar
M. Z. Xu and K. Cheng, “Convergence for sums of independent, identically distributed random variables under sublinear expectations,” 2021, https://arxiv.org/abs/2104.10430v1.
View at: Google Scholar
S. m. Hossenni and A. Nezakati, “Complete monent convergence for the dependent linear processes with random coefficients,” Acta Mathematica Sinica, English Series, vol. 35, no. 8, pp. 113–132, 2019.
View at: Google Scholar
Y. S. Chow, “On the rate of moment convergence of sample sums and extremes,” Bulletin of the Institute of Mathematics Academia Sinica, vol. 16, no. 3, pp. 177–201, 1988.
View at: Google Scholar
L. Zhang and J. Lin, “Marcinkiewicz’s strong law of large numbers for nonlinear expectations,” Statistics & Probability Letters, vol. 137, pp. 269–276, 2018.
View at: Publisher Site | Google Scholar
R. Hu and Q. Y. Wu, “Complete convergence for weighted sums of widely acceptable random variables under sublinear expectations,” Discrete Dynamics in Nature and Society, vol. 2021, Article ID 5526609, 10 pages, 2021.
View at: Publisher Site | Google Scholar
M. L. Guo, “Equivalent conditions of complete convergence for weighted sums of sequences of negatively dependent raodom variables,” Abstract and Applied Analysis, vol. 2012, Article ID 425969, 13 pages, 2012.
View at: Google Scholar

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Copyright © 2021 Mingzhou Xu and Kun Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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