| Title:
|
Some mean convergence and complete convergence theorems for sequences of $m$-linearly negative quadrant dependent random variables (English) |
| Author:
|
Wu, Yongfeng |
| Author:
|
Rosalsky, Andrew |
| Author:
|
Volodin, Andrei |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
58 |
| Issue:
|
5 |
| Year:
|
2013 |
| Pages:
|
511-529 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The structure of linearly negative quadrant dependent random variables is extended by introducing the structure of $m$-linearly negative quadrant dependent random variables ($m=1,2,\dots $). For a sequence of $m$-linearly negative quadrant dependent random variables $\{X_n, n\ge 1\}$ and $1<p<2$ (resp. $1\le p <2$), conditions are provided under which $n^{-1/p} \sum _{k=1}^{n} (X_k - EX_k) \to 0$ in $L^1$ (resp. in $L^p$). Moreover, for $1\le p < 2$, conditions are provided under which $n^{-1/p} \sum _{k=1}^{n} (X_k - EX_k)$ converges completely to $0$. The current work extends some results of Pyke and Root (1968) and it extends and improves some results of Wu, Wang, and Wu (2006). An open problem is posed. (English) |
| Keyword:
|
$m$-linearly negative quadrant dependence |
| Keyword:
|
mean convergence |
| Keyword:
|
complete convergence |
| MSC:
|
60F15 |
| MSC:
|
60F25 |
| idZBL:
|
Zbl 06282094 |
| idMR:
|
MR3104616 |
| DOI:
|
10.1007/s10492-013-0030-6 |
| . |
| Date available:
|
2013-09-14T11:40:52Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143430 |
| . |
| Reference:
|
[1] Chandra, T. K.: Uniform integrability in the Cesàro sense and the weak law of large numbers.Sankhyā, Ser. A 51 (1989), 309-317. Zbl 0721.60024, MR 1175608 |
| Reference:
|
[2] Fuk, D. H., Nagaev, S. V.: Probability inequalities for sums of independent random variables.Theory Probab. Appl. 16 (1971), 643-660. Zbl 0259.60024, MR 0293695 |
| Reference:
|
[3] Hsu, P. L., Robbins, H.: Complete convergence and the law of large numbers.Proc. Natl. Acad. Sci. USA 33 (1947), 25-31. Zbl 0030.20101, MR 0019852, 10.1073/pnas.33.2.25 |
| Reference:
|
[4] Joag-Dev, K., Proschan, F.: Negative association of random variables, with applications.Ann. Stat. 11 (1983), 286-295. Zbl 0508.62041, MR 0684886, 10.1214/aos/1176346079 |
| Reference:
|
[5] Ko, M.-H., Choi, Y.-K., Choi, Y.-S.: Exponential probability inequality for linearly negative quadrant dependent random variables.Commun. Korean Math. Soc. 22 (2007), 137-143. Zbl 1168.60336, MR 2286902, 10.4134/CKMS.2007.22.1.137 |
| Reference:
|
[6] Ko, M.-H., Ryu, D.-H., Kim, T.-S.: Limiting behaviors of weighted sums for linearly negative quadrant dependent random variables.Taiwanese J. Math. 11 (2007), 511-522. Zbl 1126.60026, MR 2333362, 10.11650/twjm/1500404705 |
| Reference:
|
[7] Lehmann, E. L.: Some concepts of dependence.Ann. Math. Stat. 37 (1966), 1137-1153. Zbl 0146.40601, MR 0202228, 10.1214/aoms/1177699260 |
| Reference:
|
[8] Newman, C. M.: Asymptotic independence and limit theorems for positively and negatively dependent random variables.Inequalities in Statistics and Probability. IMS Lecture Notes Monogr. Ser. 5 Y. L. Tong Inst. Math. Statist. Hayward (1984), 127-140. MR 0789244 |
| Reference:
|
[9] Cabrera, M. Ordóñez, Volodin, A. I.: Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability.J. Math. Anal. Appl. 305 (2005), 644-658. MR 2131528, 10.1016/j.jmaa.2004.12.025 |
| Reference:
|
[10] Pyke, R., Root, D.: On convergence in $r$-mean of normalized partial sums.Ann. Math. Stat. 39 (1968), 379-381. Zbl 0164.47303, MR 0224137, 10.1214/aoms/1177698400 |
| Reference:
|
[11] Sung, S. H., Lisawadi, S., Volodin, A.: Weak laws of large numbers for arrays under a condition of uniform integrability.J. Korean Math. Soc. 45 (2008), 289-300. Zbl 1136.60319, MR 2375136, 10.4134/JKMS.2008.45.1.289 |
| Reference:
|
[12] Wan, C. G.: Law of large numbers and complete convergence for pairwise NQD random sequences.Acta Math. Appl. Sin. 28 (2005), 253-261 Chinese. MR 2157985 |
| Reference:
|
[13] Wang, J. F., Zhang, L. X.: A Berry-Esseen theorem for weakly negatively dependent random variables and its applications.Acta Math. Hung. 110 (2006), 293-308. Zbl 1121.60024, MR 2213231, 10.1007/s10474-006-0024-x |
| Reference:
|
[14] Wang, X., Hu, S., Yang, W., Li, X.: Exponential inequalities and complete convergence for a LNQD sequence.J. Korean Statist. Soc. 39 (2010), 555-564. Zbl 1294.60037, MR 2780225, 10.1016/j.jkss.2010.01.002 |
| Reference:
|
[15] Wu, Q., Wang, Y., Wu, Y.: On some limit theorems for sums of NA random matrix sequences.Chin. J. Appl. Probab. Stat. 22 (2006), 56-62 Chinese. Zbl 1167.60315, MR 2275261 |
| . |
