Let $ \{Y_n, n\geq 1\} $ be sequence of random variables with $ EY_n = 0 $ and $ \sup_nE|Y_n|^p < \infty $ for each $ p > 2 $ satisfying Rosenthal type inequality. In this paper, the law of the iterated logarithm for a class of random variable sequence with non-identical distributions is established by the Rosenthal type inequality and Berry-Esseen bounds. The results extend the known ones from i.i.d and NA cases to a class of random variable satisfying Rosenthal type inequality.
Citation: Haichao Yu, Yong Zhang. The law of iterated logarithm for a class of random variables satisfying Rosenthal type inequality[J]. AIMS Mathematics, 2021, 6(10): 11076-11083. doi: 10.3934/math.2021642
Let $ \{Y_n, n\geq 1\} $ be sequence of random variables with $ EY_n = 0 $ and $ \sup_nE|Y_n|^p < \infty $ for each $ p > 2 $ satisfying Rosenthal type inequality. In this paper, the law of the iterated logarithm for a class of random variable sequence with non-identical distributions is established by the Rosenthal type inequality and Berry-Esseen bounds. The results extend the known ones from i.i.d and NA cases to a class of random variable satisfying Rosenthal type inequality.
| [1] |
M. Peligrad, Convergence rates of the strong law for stationary mixing sequences, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 70 (1985), 307-314. doi: 10.1007/BF02451434
|
| [2] |
X. Zhou, Complete moment convergence of moving average processes under $\varphi$-mixing assumptions, Stat. Probabil. Lett., 80 (2010), 285-292. doi: 10.1016/j.spl.2009.10.018
|
| [3] |
J. F. Wang, F. B. Lu, Inequalities of maximum of partial sums and weak convergence for a class of weak dependent random variables, Acta. Math. Sin., 22 (2006), 693-700. doi: 10.1007/s10114-005-0601-x
|
| [4] |
S. Utev, M. Peligrad, Maximal inequalities and an invariance principle for a class of weakly dependent random variables, J. Theor. Probab., 16 (2003), 101-115. doi: 10.1023/A:1022278404634
|
| [5] |
Q. M. Shao, A comparison theorem on moment inequalities between negatively associated and independent random variables, J. Theor. Probab., 13 (2000), 343-356. doi: 10.1023/A:1007849609234
|
| [6] |
G. Stoica, A note on the rate of convergence in the strong law of large numbers for martingales, J. Math. Anal. Appl., 381 (2011), 910-913. doi: 10.1016/j.jmaa.2011.04.008
|
| [7] |
A. T. Shen, Probability inequalities for END sequence and their applications, J. Inequal. Appl., 2011 (2011), 98. doi: 10.1186/1029-242X-2011-98
|
| [8] |
D. M. Yuan, J. An, Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications, Sci. China Ser. A, 52 (2009), 1887-1904. doi: 10.1007/s11425-009-0154-z
|
| [9] |
A. T. Shen, Y. Zhang, A. Volodin, Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables, Metrika, 78 (2015), 295-311. doi: 10.1007/s00184-014-0503-y
|
| [10] | F. Merlev$\acute{e}$de, M. Peligrad, Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples, Ann. Probab., 41 (2013), 914-960. |
| [11] | V. V. Petrov, Limit theorems of probability theory: Sequences of independent random variables, Oxford: Oxford Science Publications, 1995. |
| [12] |
G. H. Cai, H. Wu, Law of iterated logarithm for NA sequences with non-identical distributions, Proc. Math. Sci., 117 (2007), 213-218. doi: 10.1007/s12044-007-0017-x
|
| [13] |
R. Wittmann, A general law of iterated logarithm, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 68 (1985), 521-543. doi: 10.1007/BF00535343
|
| [14] | S. Kochen, C. Stone, A note on the Borel-Cantelli lemma, Illinois J. Math., 8 (1964), 248-251. |
Haichao Yu, Yong Zhang. The law of iterated logarithm for a class of random variables satisfying Rosenthal type inequality[J]. AIMS Mathematics, 2021, 6(10): 11076-11083. doi: 10.3934/math.2021642
DownLoad: