Complete moment convergence for the dependent linear processes with application to the state observers of linear-time-invariant systems. (English) Zbl 1458.93235
Theory Probab. Appl. 65, No. 4, 570-587 (2021) and Teor. Veroyatn. Primen. 65, No. 4, (2020).
Summary: Let \(X_t=\sum_{j=-\infty}^{\infty}A_j\varepsilon_{t-j}\) be a dependent linear process, where the \(\{\varepsilon_n,\, n\in{Z}\}\) is a sequence of zero mean \(m\)-extended negatively dependent \((m\)-END, for short) random variables which is stochastically dominated by a random variable \(\varepsilon \), and \(\{A_n,\, n\in{Z}\}\) is also a sequence of zero mean \(m\)-END random variables. Under some suitable conditions, the complete moment convergence for the dependent linear processes is established. In particular, the sufficient conditions of the complete moment convergence are provided. As an application, we further study the convergence of the state observers of linear-time-invariant systems.
Keywords:
complete moment convergence; END random variables; linear processes; linear-time-invariant systemsReferences:
| [1] | A. Adler and A. Rosalsky, Some general strong laws for weighted sums of stochastically dominated random variables, Stochastic Anal. Appl., 5 (1987), pp. 1-16, https://doi.org/10.1080/07362998708809104. · Zbl 0617.60028 |
| [2] | A. Adler, A. Rosalsky, and R. L. Taylor, Strong laws of large numbers for weighted sums of random elements in normed linear spaces, Internat. J. Math. Math. Sci., 12 (1989), pp. 507-529, https://doi.org/10.1155/S0161171289000657. · Zbl 0679.60009 |
| [3] | Z. Bai and C. Su, The complete convergence for partial sums of i.i.d. random variables, Sci. Sinica Ser. A, 28 (1985), pp. 1261-1277. · Zbl 0554.60039 |
| [4] | V. M. Charitopoulos and V. Dua, A unified framework for model-based multi-objective linear process and energy optimisation under uncertainty, Appl. Energy, 186, Part 3 (2017), pp. 539-548, https://doi.org/10.1016/j.apenergy.2016.05.082. |
| [5] | P. Chen, P. Bai, and S. H. Sung, The von Bahr-Esseen moment inequality for pairwise independent random variables and applications, J. Math. Anal. Appl., 419 (2014), pp. 1290-1302, https://doi.org/10.1016/j.jmaa.2014.05.067. · Zbl 1316.60028 |
| [6] | Y. S. Chow, On the rate of moment convergence of sample sums and extremes, Bull. Inst. Math. Acad. Sinica, 16 (1988), pp. 177-201. · Zbl 0655.60028 |
| [7] | S. M. Hosseini and A. Nezakati, Complete moment convergence for the dependent linear processes with random coefficients, Acta Math. Sin. (Engl. Ser.), 35 (2019), pp. 1321-1333, https://doi.org/10.1007/s10114-019-8205-z. · Zbl 1420.60041 |
| [8] | P. L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Natl. Acad. Sci. USA, 33 (1947), pp. 25-31, https://doi.org/10.1073/pnas.33.2.25. · Zbl 0030.20101 |
| [9] | T.-C. Hu, K.-L. Wang, and A. Rosalsky, Complete convergence theorems for extended negatively dependent random variables, Sankhyā A, 77 (2015), pp. 1-29, https://doi.org/10.1007/s13171-014-0058-z. · Zbl 1317.60027 |
| [10] | M.-H. Ko, Strong laws of large numbers for linear processes generated by associated random variables in a Hilbert space, Honam Math. J., 30 (2008), pp. 703-711, https://doi.org/10.5831/HMJ.2008.30.4.703. · Zbl 1180.60027 |
| [11] | M.-H. Ko, The complete moment convergence for CNA random vectors in Hilbert spaces, J. Inequal. Appl., 2017 (2017), 290, https://doi.org/10.1186/s13660-017-1566-x. · Zbl 1375.60072 |
| [12] | M.-H. Ko, On complete moment convergence for CAANA random vectors in Hilbert spaces, Statist. Probab. Lett., 138 (2018), pp. 104-110, https://doi.org/10.1016/j.spl.2018.02.068. · Zbl 1391.60054 |
| [13] | E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist., 37 (1966), pp. 1137-1153, https://doi.org/10.1214/aoms/1177699260. · Zbl 0146.40601 |
| [14] | H.-Y. Liang, Complete convergence for weighted sums of negatively associated random variables, Statist. Probab. Lett., 48 (2000), pp. 317-325, https://doi.org/10.1016/S0167-7152(00)00002-X. · Zbl 0960.60027 |
| [15] | L. Liu, Precise large deviations for dependent random variables with heavy tails, Statist. Probab. Lett., 79 (2009), pp. 1290-1298, https://doi.org/10.1016/j.spl.2009.02.001. · Zbl 1163.60012 |
| [16] | H. Naderi, M. Amini, and A. Bozorgnia, On the rate of complete convergence for weighted sums of NSD random variables and an application, Appl. Math. J. Chinese Univ. Ser. B, 32 (2017), pp. 270-280, https://doi.org/10.1007/s11766-017-3437-0. · Zbl 1399.60053 |
| [17] | M. A. Nowak, F. Michor, and Y. Iwasa, The linear process of somatic evolution, Proc. Natl. Acad. Sci. USA, 100 (2003), pp. 14966-14969, https://doi.org/10.1073/pnas.2535419100. |
| [18] | K. Ogata, Modern Control Engineering, 4th ed., Prentice-Hall, Englewood Cliffs, NJ, 2002. · Zbl 0756.93060 |
| [19] | R. I. A. Patterson, J. Singh, M. Balthasar, M. Kraft, and J. R. Norris, The linear process deferment algorithm: A new technique for solving population balance equations, SIAM J. Sci. Comput., 28 (2006), pp. 303-320, https://doi.org/10.1137/040618953. · Zbl 1105.65302 |
| [20] | D. H. Qiu and P. Y. Chen, Complete and complete moment convergence for weighted sums of widely orthant dependent random variables, Acta Math. Sin. (Engl. Ser.), 30 (2014), pp. 1539-1548, https://doi.org/10.1007/s10114-014-3483-y. · Zbl 1296.60075 |
| [21] | E. Seneta, Regularly Varying Functions, Lecture Notes in Math. 508, Springer-Verlag, Berlin, 1976, https://doi.org/10.1007/BFb0079658. · Zbl 0324.26002 |
| [22] | A. Shen, M. Xue, and A. Volodin, Complete moment convergence for arrays of rowwise NSD random variables, Stochastics, 88 (2016), pp. 606-621, https://doi.org/10.1080/17442508.2015.1110153. · Zbl 1337.60038 |
| [23] | A. Shen, Y. Zhang, and W. Wang, Complete convergence and complete moment convergence for extended negatively dependent random variables, Filomat, 31 (2017), pp. 1381-1394, https://doi.org/10.2298/FIL1705381S. · Zbl 1488.60059 |
| [24] | R. Shibata, Asymptotically efficient selection of the order of the model for estimating parameters of a linear process, Ann. Statist., 8 (1980), pp. 147-164, https://doi.org/10.1214/aos/1176344897. · Zbl 0425.62069 |
| [25] | L. V. Thanh, G. G. Yin, and L. Y. Wang, State observers with random sampling times and convergence analysis of double-indexed and randomly weighted sums of mixing processes, SIAM J. Control Optim., 49 (2011), pp. 106-124, https://doi.org/10.1137/10078671X. · Zbl 1219.93139 |
| [26] | J. Wang and Q. Wu, Central limit theorem for stationary linear processes generated by linearly negative quadrant-dependent sequence, J. Inequal. Appl., 2012 (2012), 45, https://doi.org/10.1186/1029-242X-2012-45. · Zbl 1450.60022 |
| [27] | L. Y. Wang, C. Li, G. G. Yin, L. Guo, and C.-Z. Xu, State observability and observers of linear-time-invariant systems under irregular sampling and sensor limitations, IEEE Trans. Automat. Control, 56 (2011), pp. 2639-2654, https://doi.org/10.1109/TAC.2011.2122570. · Zbl 1368.93062 |
| [28] | X. Wang, T.-C. Hu, A. Volodin, and S. Hu, Complete convergence for weighted sums and arrays of rowwise extended negatively dependent random variables, Comm. Statist. Theory Methods, 42 (2013), pp. 2391-2401, https://doi.org/10.1080/03610926.2011.609321. · Zbl 1276.60036 |
| [29] | X. Wang, Y. Wu, and S. Hu, Exponential probability inequality for \(m\)-END random variables and its applications, Metrika, 79 (2016), pp. 127-147, https://doi.org/10.1007/s00184-015-0547-7. · Zbl 1330.60031 |
| [30] | X. Wang, Y. Wu, and S. Hu, Complete moment convergence for double-indexed randomly weighted sums and its applications, Statistics, 52 (2018), pp. 503-518, https://doi.org/10.1080/02331888.2018.1436548. · Zbl 1391.60055 |
| [31] | Y. Wu, X. Wang, and S. Hu, Complete moment convergence for weighted sums of weakly dependent random variables and its application in nonparametric regression model, Statist. Probab. Lett., 127 (2017), pp. 56-66, https://doi.org/10.1016/j.spl.2017.03.027. · Zbl 1378.60063 |
| [32] | Y. Wu, On complete moment convergence for arrays of rowwise negatively associated random variables, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 108 (2014), pp. 669-681, https://doi.org/10.1007/s13398-013-0133-7. · Zbl 1296.60078 |
| [33] | G. Zhang, Complete convergence for Sung’s type weighted sums of END random variables, J. Inequal. Appl., 2014 (2014), 353, https://doi.org/10.1186/1029-242X-2014-353. · Zbl 1323.60054 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
