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Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications. (English) Zbl 1184.62099
A sequence of random variables \(\{X_i, 1 \leq i \leq n \}\) is called negatively asociated (NA) if for every pair of disjoint subsets \( A \) and \( B\) of \(\{ 1, 2, \dots , n \}\),
\[ \text{Cov}(f(X_i, i \in A), g(X_j , j \in B)) \leq 0, \] whenever \(f\) and \(g\) are coordinatewise nondecreasing and the covariance exists. A sequence of random variables \(\{ X_n\), \(n\geq 1\}\) is called asymptotically almost negatively associated (AANA) if there exists a nonnegative sequence \(q(n) \rightarrow 0 \) as \( n \rightarrow \infty\) such that
\[ \text{Cov}(f(X_n), g(X_{n+1}, \dots , X_{n+k} )) \leq q(n)(\text{Var}(f(X_n)) \text{Var}(g(X_{n+1}, \dots ,X_{n+k})))^{1/2}, \] for all \( n, k \geq 1 \) and for all coordinatewise nondecreasing continuous functions \(f\) and \(g\) whenever the variances exist. For NA random variables a lot of sharp and elegant estimates are available. Some Rosenthal type moment inequalities are also introduced. For AANA random variables, some excellent results are also available. However, for AANA random variables, Rosenthal type inequalities are not yet available.
The authors establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively associated random variables, which extend the corresponding results for negatively associated random variables. As application of these inequalities, by employing the notions of residual Cesàro \(\alpha\)-integrability and strong residual Cesàro \(\alpha\)-integrability, they derive some results on \(L_p\)-convergence, where \(1 < p < 2\), and on complete convergence. In addition, they estimate the rate of convergence in the Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.

62H20 Measures of association (correlation, canonical correlation, etc.)
60F15 Strong limit theorems
60E15 Inequalities; stochastic orderings
62H05 Characterization and structure theory for multivariate probability distributions; copulas
Full Text: DOI
[1] Joag-Dev K, Proschan F. Negative association of random variables with applications. Ann Statist, 11: 286–295: (1983) · Zbl 0508.62041 · doi:10.1214/aos/1176346079
[2] Block H W, Savits T H, Shaked M. Some concepts of negative dependence. Ann Probab, 10: 765–772: (1982) · Zbl 0501.62037 · doi:10.1214/aop/1176993784
[3] Matula P. A note on the almost sure convergence of sums of negatively dependent random variables. Statist Probab Lett, 15: 209–213: (1992) · Zbl 0925.60024 · doi:10.1016/0167-7152(92)90191-7
[4] Chandra T K, Ghosal S. Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables. Acta Math Hung, 71: 327–336: (1996) · Zbl 0853.60032 · doi:10.1007/BF00114421
[5] Chandra T K, Ghosal S. The strong law of large numbers for weighted averages under dependence assumptions. J Theoret Probab, 9: 797–809: (1996) · Zbl 0857.60021 · doi:10.1007/BF02214087
[6] Su C, Zhao L C, Wang Y B. Moment inequalities and weak convergence for negatively associated sequences. Sci China Ser A, 40: 172–182: (1997) · Zbl 0907.60023 · doi:10.1007/BF02874436
[7] Shao Q M, Su C. The law of the iterated logarithm for negatively associated random variables. Stoch Proc Appl, 83: 139–148: (1999) · Zbl 0997.60023 · doi:10.1016/S0304-4149(99)00026-5
[8] Shao Q M. A comparison theorem on maximal inequalities between negatively associated and independent random variables. J Theort Probab, 13: 343–356: (2000) · Zbl 0971.60015 · doi:10.1023/A:1007849609234
[9] Ko M H, Kim T S, Lin Z Y. The Hájek-Rènyi inequality for the AANA random variables and its applications. Taiwanese J Math, 9: 111–122: (2005) · Zbl 1069.60022
[10] Wang Y B, Yan J G, Cheng F Y. The strong law of large numbers and the iterated logarithm for product sums of NA and AANA random variables. Southeast Asian Bull Math, 27: 369–384: (2003) · Zbl 1061.60031
[11] Baum E, Katz M. Convergence rates in the law of large numbers. Trans Amer Math Soc, 120: 108–123: (1965) · Zbl 0142.14802 · doi:10.1090/S0002-9947-1965-0198524-1
[12] Wang J F, Lu F B. Inequalities of maximum of partial sums and weak convergence for a class of weak dependent random variables. Acta Math Sin Engl Ser, 22: 693–700: (2006) · Zbl 1102.60023 · doi:10.1007/s10114-005-0601-x
[13] Zhang L X. A functional central limit theorem for asymptotically negatively dependent random fields. Acta Math Hung, 86: 237–259: (2000) · Zbl 0964.60035 · doi:10.1023/A:1006720512467
[14] Chow Y S. On the L p-convergence for n /p S n, 0 < p < 2. Ann Math Stat, 42: 393–394: (1971) · Zbl 0235.60031 · doi:10.1214/aoms/1177693530
[15] Bose A, Chandra T K. Cesàro uniform integrability and L p-convergence. Sankhyā Ser A, 55: 12–28: (1993) · Zbl 0809.60043
[16] Chandra T K, Goswami A. Cesàro \(\alpha\)-integrability and laws of large numbers II. J Theoret Probab, 19: 789–816: (2006) · Zbl 1111.60018 · doi:10.1007/s10959-006-0038-x
[17] Landers D, Rogge L. Laws of large numbers for uncorrelated Cesàro uniformly integrable random variables. Sankhyā Ser A, 59: 301.310: (1997) · Zbl 0953.60010
[18] Peligrad M, Gut A. Almost sure results for a class of dependent random variables. J Theoret Probab, 12: 87–104: (1999) · Zbl 0928.60025 · doi:10.1023/A:1021744626773
[19] Peligrad M. Convergence rates of strong law for stationary mixing sequences. Z Wahrsch Verw Geb, 70: 307–314: (1985) · Zbl 0554.60038 · doi:10.1007/BF02451434
[20] Etemadi, N. An elementary proof of the strong law of large numbers. Z Wahrsch Verw Geb, 55: 119–122: (1981) · Zbl 0438.60027 · doi:10.1007/BF01013465
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