# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Inequalities of maximum of partial sums and weak convergence for a class of weak dependent random variables. (English) Zbl 1102.60023
Summary: We establish a Rosenthal-type inequality of the maximum of partial sums for ${\rho }^{-}$-mixing random fields. As its applications we get the Hájek-Rényi inequality and weak convergence of sums of ${\rho }^{-}$-mixing sequence. These results extend related results for NA sequences and ${\rho }^{*}$-mixing random fields.
##### MSC:
 60F05 Central limit and other weak theorems 60E15 Inequalities in probability theory; stochastic orderings
##### References:
 [1] Zhang, L. X., Wang, X. Y.: Convergence rates in the strong laws of asymptotically negatively associated random fields. Appl. Math. J. Chinese Univ., 14(4), 406–416 (1999) · Zbl 0952.60031 · doi:10.1007/s11766-999-0070-6 [2] Zhang, L. X.: A functional central limit theorem for asymptotically negatively dependent random fields. Acta Math. Hungar., 86(3), 237–259 (2000) · Zbl 0964.60035 · doi:10.1023/A:1006720512467 [3] Zhang, L. X.: Central limit theorems for asymptotically negatively associated random fields. Acta Math. Sinica, English Series., 16(4), 691–710 (2000) · Zbl 0977.60020 · doi:10.1007/s101140000084 [4] Joag Dev, K., Proschan, F.: Negative Association of Random Variables with Applications. Ann. Statist., 11, 268–295 (1983) [5] Bradley, R. C.: On the spectral density and asymptotic normality of weakly dependent random fields. J. Theoret. Probab., 5, 355–373 (1992) · Zbl 0787.60059 · doi:10.1007/BF01046741 [6] Peligrad, M., Gut, A.: Almost-sure results for a class of dependent random variables. J. Theoret. Probab., 12(1), 87–103 (1999) · Zbl 0928.60025 · doi:10.1023/A:1021744626773 [7] Utev, S., Peligrad, M.: Maximal inequalities and an invariance principle for a class of weakly dependent random variables. J. Theoret. Probab., 16(1), 101–115 (2003) · Zbl 1012.60022 · doi:10.1023/A:1022278404634 [8] Zhang, L. X.: Convergence rates in the strong laws of nonstationary $\rho$*ixing random fields. Acta Math. Scientia , Ser. B, 20, 303–312 (2000) (in Chinese) [9] Matula, P.: A note on the almost sure convergence of negatively deoendent variables. Statist. Probab. Lett., 15, 209–213 (1992) · Zbl 0925.60024 · doi:10.1016/0167-7152(92)90191-7 [10] Liu, J. J., Gan, S. X., Chen, P. Y.: The Hájeck–Rányi inequality for the NA random variables and its application. Statist. Probab. Lett., 99–105 (1999) [11] Zhang, L. X., Wen, J. W.: A weak convergence for negatively associated fields. Statist. Probab. Lett., 53, 259–267 (2001) · Zbl 0994.60026 · doi:10.1016/S0167-7152(01)00021-9 [12] Su, C., Zhao, L. C., Wang, Y. B.: The moment inequality for the NA random variables and the weak convergence. Sci. China, Ser. A, 26, 1091–1099 (in Chinese) [13] Patrick, B.: Convergence of Probability Measures, Wiley, New York, 1999