Limiting behaviors of linear processes with random coefficients based on \(m\)-ANA random variables. (English) Zbl 07565746
The authors begin with the definition of negatively associated (NA) random variables in a finite family of random variables \(X_1,\dots,X_n\) by using the negativity of Covariances between suitable functions of \(X_i\) and \(X_j\) where \(i,j \in \{1,2,\dots,n\}\). Then they define asymptotic negatively associated (ANA) introduced by Zhang and Wang (see the authors’ reference) by suitable asymptotic decay. Then they extend the concept of ANA random variables to the concept of m-ANA random variables (see [Y. Wu et al., Stat. Pap. 62, No. 5, 2169–2194 (2021; Zbl 1479.60066)] in the authors’ reference). In this paper the authors investigate the complete convergence and complete moment convergence of linear processes with random coefficients based on m-ANA random variables. The authors provide some generalizations of earlier results in the literature. Some applications to certain strong law of large numbers are given as corollaries for linear processes of m-ANA random variables with random coefficients.
Reviewer: Rasul A. Khan (Solon)
Keywords:
linear process; random coefficients; \(m\)-ANA random variables; complete convergence; complete moment convergence; strong law of large numbersCitations:
Zbl 1479.60066References:
| [1] | A. ADLER, A. ROSALSKY,Some general strong laws for weighted sums of stochastically dominated random variables, Stochastic Analysis & Applications, 5 (1), 1-16, 1987. · Zbl 0617.60028 |
| [2] | A. ANDRE, R. ANDREW, R. L. TAYLOR,Strong laws of large numbers for weighted sums of random elements in normed linear spaces, International Journal of Mathematics and Mathematical Sciences, 12 (3), 507-529, 1989. · Zbl 0679.60009 |
| [3] | J. I. BAEK, T. S. KIM, H. Y. LIANG,On the convergence of moving average processes under dependent conditions, Australian & New Zealand Journal of Statistics, 45 (3), 331-342, 2003. · Zbl 1082.60028 |
| [4] | R. M. BURTON, H. DEHLING,Moderate deviations for some weakly dependent random processes, Statistics & Probability Letters, 9 (5), 397-401, 1990. · Zbl 0699.60016 |
| [5] | K. BUDSABA, P. Y. CHEN, A. VOLODIN,Limiting behavior of moving average processes based on a sequence ofρ−mixing random variables, Thailand Statistician, 5, 69-80, 2007a. · Zbl 1143.62050 |
| [6] | K. BUDSABA, P. Y. CHEN, A. VOLODIN,Limiting behaviour of moving average processes based on a sequence ofρ−mixing and negatively associated random variables, Lobachevskii Journal of Mathematics, 26, 17-25, 2007b. · Zbl 1132.60028 |
| [7] | P. Y. CHEN, T.-C. HU, A. VOLODIN,Limiting behaviour of moving average processes under negative association assumption, Theory of Probability & Mathematical Statistics, 77, 165-176, 2008. · Zbl 1199.60074 |
| [8] | P. Y. CHEN, T.-C. HU, A. VOLODIN,Limiting behaviour of moving average processes underϕmixing assumption, Statistics & Probability Letters, 79 (1), 105-111, 2009. · Zbl 1154.60026 |
| [9] | Y. S. CHOW,On the rate of moment convergence of sample sums and extremes, Bulletin of the Institute of Mathematics, Academia Sinica, 16 (3), 177-201, 1988. · Zbl 0655.60028 |
| [10] | S. M. HOSSEINI, A. NEZAKATI,Complete moment convergence for the dependent linear processes with random coefficients, Acta Mathematica Sinica, English Series, 35, 1321-1333, 2019. · Zbl 1420.60041 |
| [11] | S. M. HOSSEINI, A. NEZAKATI,Convergence rates in the law of large numbers for END linear processes with random coefficients, Communications in Statistics-Theory and Methods, 49 (1), 88- 98, 2020. · Zbl 07549021 |
| [12] | P. L. HSU, H. ROBBINS,Complete convergence and the law of large numbers, Proceedings of the National Academy of Sciences U.S.A., 33, 25-31, 1947. · Zbl 0030.20101 |
| [13] | I. A. IBRAGIMOV,Some limit theorems for stationary processes, Theory of Probability and Its Applications, 7 (4), 361-392, 1962. · Zbl 0119.14204 |
| [14] | T. S. KIM, M. H. KO,Complete moment convergence of moving average processes under dependence assumptions, Statistics & Probability Letters, 78 (7), 839-846, 2008. · Zbl 1140.60315 |
| [15] | M. H. KO, T. S. KIM, D. H. RYU,On the complete moment convergence of moving average processes generated byρ∗-mixing sequences, Communications of the Korean Mathematical Society, 23 (4), 597-606, 2008. · Zbl 1168.60346 |
| [16] | R. KULIK,Limit theorems for moving averages with random coefficients and heavy-tailed noise, Journal of Applied Probability, 43 (1), 245-256, 2006. · Zbl 1097.62082 |
| [17] | D. LI, M. B. RAO, X. C. WANG,Complete convergence of moving average processes, Statistics & Probability Letters, 14, 111-114, 1992. · Zbl 0756.60031 |
| [18] | H. Y. LIANG, T. S. KIM, J. I. BAEK,On the convergence of moving average processes under negatively associated random variables, Indian Journal of Pure and Applied Mathematics, 34 (3), 461-476, 2003. · Zbl 1030.60017 |
| [19] | D. W. LU, L. N. WANG,Complete moment convergence for the widely orthant dependent linear processes with random coefficients, Communications in Statistics-Theory and Methods, 51 (3), 811- 825, 2022. · Zbl 07532305 |
| [20] | P. SAAVEDRA, C. N. HERNANDEZ´, I. LUENGO, J. ARTILES, A. SANTANA,Estimation of population spectrum for linear processes with random coefficients, Computational Statistics, 23 (1), 79-98, 2008. |
| [21] | Y. WU, X. J. WANG, S. H. HU,Complete moment convergence for weighted sums of weakly dependent random variables and its application in nonparametric regression model, Statistics & Probability Letters, 127, 56-66, 2017. · Zbl 1378.60063 |
| [22] | Y. WU, X. J. WANG, A. T. SHEN,Strong convergence properties for weighted sums of m -asymptotic negatively associated random variables and statistical applications, Statistical Papers, 62, 2169-2194, 2021. · Zbl 1479.60066 |
| [23] | L. X. ZHANG,Complete convergence of moving average processes under dependence assumptions, Statistics & Probability Letters, 30, 165-170, 1996. · Zbl 0873.60019 |
| [24] | X. C. ZHOU,Complete moment convergence of moving average processes underϕ-mixing assumptions, Statistics & Probability Letters, 80, 285-292, 2010 · Zbl 1186.60031 |
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.
