Previous |  Up |  Next


the Doob inequality; strong law of large numbers; martingale difference array; Banach space
We establish the Doob inequality for martingale difference arrays and provide a sufficient condition so that the strong law of large numbers would hold for an arbitrary array of random elements without imposing any geometric condition on the Banach space. Some corollaries are derived from the main results, they are more general than some well-known ones.
[1] P. Assouad: Espaces $p$-lisses et $q$-convexes, inégalités de Burkholder. Séminaire Maurey-Schwartz 1975. MR 0407963 | Zbl 0318.46023
[2] R. Cairoli, J. B. Walsh: Stochastic integrals in the plane. Acta Math. 134 (1975), 111-183. DOI 10.1007/BF02392100 | MR 0420845 | Zbl 0334.60026
[3] A. Gut: Probability: A Graduate Course. Springer, New York 2005. MR 2125120 | Zbl 1151.60300
[4] J. O. Howell, R. L. Taylor: Marcinkiewicz-Zygmund Weak Laws of Large Numbers for Unconditional Random Elements in Banach Spaces, Probability in Banach Spaces. Springer, Berlin - New York 1981. MR 0647964
[5] N. V. Huan, N. V. Quang, A. Volodin: Strong laws for blockwise martingale difference arrays in Banach spaces. Lobachevskii J. Math. 31 (2010), 326-335. DOI 10.1134/S1995080210040037 | MR 2745742
[6] H. C. Kim: The Hájek-Rényi inequality for weighted sums of negatively orthant dependent random variables. Internat. J. Contemporary Math. Sci. 1 (2006), 297-303. MR 2289035
[7] Z. A. Lagodowski: Strong laws of large numbers for $\mathbb B$-valued random fields. Discrete Dynamics in Nature and Society (2009). MR 2504837
[8] F. Móricz: Strong limit theorems for quasi-orthogonal random fields. J. Multivariate Anal. 30 (1989), 255-278. DOI 10.1016/0047-259X(89)90039-0 | MR 1015372 | Zbl 0798.60033
[9] F. Móricz, U. Stadtmüller, M. Thalmaier: Strong laws for blockwise $\mathcal M$-dependent random fields. J. Theoret. Probab. 21 (2008), 660-671. DOI 10.1007/s10959-007-0127-5 | MR 2425363
[10] F. Móricz, K. L. Su, R. L. Taylor: Strong laws of large numbers for arrays of orthogonal random elements in Banach spaces. Acta Math. Hungar. 65 (1994), 1-16. DOI 10.1007/BF01874465 | MR 1275656 | Zbl 0806.60002
[11] C. Noszály, T. Tómács: A general approach to strong laws of large numbers for fields of random variables. Ann. Univ. Sci. Budapest. Sect. Math. 43 (2001), 61-78. MR 1847869 | Zbl 0993.60029
[12] N. V. Quang, N. V. Huan: On the strong law of large numbers and $\mathcal L_p$-convergence for double arrays of random elements in $p$-uniformly smooth Banach spaces. Stat. Probab. Lett. 79 (2009), 1891-1899. DOI 10.1016/j.spl.2009.05.014 | MR 2567447
[13] N. V. Quang, N. V. Huan: A characterization of $p$-uniformly smooth Banach spaces and weak laws of large numbers for $d$-dimensional adapted arrays. Sankhyā 72 (2010), 344-358. DOI 10.1007/s13171-010-0020-7 | MR 2746116 | Zbl 1213.60086
[14] N. V. Quang, N. V. Huan: A Hájek-Rényi type maximal inequality and strong laws of large numbers for multidimensional arrays. J. Inequalities Appl. 2010. MR 2765284 | Zbl 1215.60022
[15] A. Rosalsky, L. V. Thanh: On almost sure and mean convergence of normed double sums of Banach space valued random elements. Stochastic Analysis and Applications 25 (2007), 895-911. DOI 10.1080/07362990701420142 | MR 2335071 | Zbl 1124.60007
[16] A. Rosalsky, L. V. Thanh: On the strong law of large numbers for sequences of blockwise independent and blockwise $p$-orthoganal random element in rademacher type $p$ banach spaces. Probab. Math. Statist. 27 (2007), 205-222. MR 2445993
[17] 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 | MR 1777538 | Zbl 0971.60015
[18] R. T. Smythe: Strong laws of large numbers for $r$-dimensional arrays of random variables. Ann. Probab. 1 (1973), 164-170. DOI 10.1214/aop/1176997031 | MR 0346881 | Zbl 0258.60026
[19] L. V. Thanh: On the strong law of large numbers for $d$-dimensional arrays of random variables. Electron. Comm. Probab. 12 (2007), 434-441. MR 2365645 | Zbl 1128.60023
[20] W. A. Woyczyński: On Marcinkiewicz-Zygmund laws of large numbers in Banach spaces and related rates of convergence. Probab. Math. Statist. 1 (1981), 117-131. MR 0626306 | Zbl 0502.60006
[21] W. A. Woyczyński: Asymptotic Behavior of Martingales in Banach Spaces, Martingale Theory in Harmonic Analysis and Banach Spaces. Springer, Berlin - New York 1982. MR 0451394
Partner of
EuDML logo