×

Convergence rates in the law of large numbers for arrays of Banach valued martingale differences. (English) Zbl 1470.60097

Summary: We study the convergence rates in the law of large numbers for arrays of Banach valued martingale differences. Under a simple moment condition, we show sufficient conditions about the complete convergence for arrays of Banach valued martingale differences; we also give a criterion about the convergence for arrays of Banach valued martingale differences. In the special case where the array of Banach valued martingale differences is the sequence of independent and identically distributed real valued random variables, our result contains the theorems of Hsu-Robbins-Erdös, Spitzer , and Baum and Katz . In the real valued single martingale case, it generalizes the results of G. Alsmeyer [Stochastic Processes Appl. 36, No. 2, 181–194 (1990; Zbl 0725.60023)]. The consideration of Banach valued martingale arrays (rather than a Banach valued single martingale) makes the results very adapted in the study of weighted sums of identically distributed Banach valued random variables, for which we prove new theorems about the rates of convergence in the law of large numbers. The results are established in a more general setting for sums of infinite many Banach valued martingale differences. The obtained results improve and extend those of S. Ghosal and T. K. Chandra [J. Theor. Probab. 11, No. 3, 621–631 (1998; Zbl 0913.60029)] .

MSC:

60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hsu, P. L.; Robbins, H., Complete convergence and the law of large numbers, Proceedings of the National Academy of Sciences of the United States of America, 33, 25-31 (1947) · Zbl 0030.20101 · doi:10.1073/pnas.33.2.25
[2] Erdös, P., On a theorem of Hsu and Robbins, Annals of Mathematical Statistics, 20, 286-291 (1949) · Zbl 0033.29001 · doi:10.1214/aoms/1177730037
[3] Erdös, P., Remark on my paper ‘On a theorem of Hsu and Robbin’, Annals of Mathematical Statistics, 21, article 138 (1950) · Zbl 0035.21403
[4] Spitzer, F., A combinatorial lemma and its application to probability theory, Transactions of the American Mathematical Society, 82, 323-339 (1956) · Zbl 0071.13003 · doi:10.1090/S0002-9947-1956-0079851-X
[5] Katz, M. L., The probability in the tail of a distribution, Annals of Mathematical Statistics, 34, 312-318 (1963) · Zbl 0209.49503 · doi:10.1214/aoms/1177704268
[6] Baum, L. E.; Katz, M., Convergence rates in the law of large numbers, Transactions of the American Mathematical Society, 120, 108-123 (1965) · Zbl 0142.14802 · doi:10.1090/S0002-9947-1965-0198524-1
[7] Lai, T. L., Limit theorems for delayed sums, Annals of Probability, 2, 432-440 (1974) · Zbl 0305.60009
[8] Gafurov, M. U.; Slastnikov, A. D., Some problems on the exit of a random walk from a curvilinear boundary, and large deviations, Theory of Probability and Its Applications, 32, 2, 327-348 (1987) · Zbl 0629.60080
[9] Li, D. L.; Rao, M. B.; Jiang, T. F.; Wang, X. C., Complete convergence and almost sure convergence of weighted sums of random variables, Journal of Theoretical Probability, 8, 1, 49-76 (1995) · Zbl 0814.60026 · doi:10.1007/BF02213454
[10] Hu, T.-C.; Szynal, D.; Volodin, A. I., A note on complete convergence for arrays, Statistics & Probability Letters, 38, 1, 27-31 (1998) · Zbl 0910.60017 · doi:10.1016/S0167-7152(98)00150-3
[11] Hu, T.-C.; Volodin, A., Addendum to: ‘A note on complete convergence for arrays‘, Statistics & Probability Letters, 38, 1, 27-31 (1998) · Zbl 0910.60017 · doi:10.1016/S0167-7152(99)00209-6
[12] Hu, T.-C.; Ordóñez Cabrera, M.; Sung, S. H.; Volodin, A., Complete convergence for arrays of rowwise independent random variables, Korean Mathematical Society, 18, 2, 375-383 (2003) · Zbl 1101.60324 · doi:10.4134/CKMS.2003.18.2.375
[13] Kuczmaszewska, A., On some conditions for complete convergence for arrays, Statistics & Probability Letters, 66, 4, 399-405 (2004) · Zbl 1074.60038 · doi:10.1016/j.spl.2003.11.010
[14] Sung, S. H.; Volodin, A. I.; Hu, T.-C., More on complete convergence for arrays, Statistics & Probability Letters, 71, 4, 303-311 (2005) · Zbl 1087.60030 · doi:10.1016/j.spl.2004.11.006
[15] Kruglov, V. M.; Volodin, A. I.; Hu, T.-C., On complete convergence for arrays, Statistics & Probability Letters, 76, 15, 1631-1640 (2006) · Zbl 1100.60014 · doi:10.1016/j.spl.2006.04.006
[16] Lesigne, E.; Volný, D., Large deviations for martingales, Stochastic Processes and their Applications, 96, 1, 143-159 (2001) · Zbl 1059.60033 · doi:10.1016/S0304-4149(01)00112-0
[17] Stoica, G., Baum-Katz-Nagaev type results for martingales, Journal of Mathematical Analysis and Applications, 336, 2, 1489-1492 (2007) · Zbl 1130.60020 · doi:10.1016/j.jmaa.2007.03.012
[18] Alsmeyer, G., Convergence rates in the law of large numbers for martingales, Stochastic Processes and their Applications, 36, 2, 181-194 (1990) · Zbl 0725.60023 · doi:10.1016/0304-4149(90)90090-F
[19] Ghosal, S.; Chandra, T. K., Complete convergence of martingale arrays, Journal of Theoretical Probability, 11, 3, 621-631 (1998) · Zbl 0913.60029 · doi:10.1023/A:1022646429754
[20] Gut, A., Complete convergence and Cesàro summation for i.i.d. random variables, Probability Theory and Related Fields, 97, 1-2, 169-178 (1993) · Zbl 0793.60034 · doi:10.1007/BF01199318
[21] Lanzinger, H.; Stadtmüller, U., Baum-Katz laws for certain weighted sums of independent and identically distributed random variables, Bernoulli, 9, 6, 985-1002 (2003) · Zbl 1047.60045 · doi:10.3150/bj/1072215198
[22] Wang, Y.; Liu, X.; Su, C., Equivalent conditions of complete convergence for independent weighted sums, Science in China A, 41, 9, 939-949 (1998) · Zbl 0922.60033 · doi:10.1007/BF02880003
[23] Yu, K. F., Complete convergence of weighted sums of martingale differences, Journal of Theoretical Probability, 3, 2, 339-347 (1990) · Zbl 0698.60035 · doi:10.1007/BF01045165
[24] Li, D. L.; Rao, M. B.; Wang, X. C., Complete convergence of moving average processes, Statistics & Probability Letters, 14, 2, 111-114 (1992) · Zbl 0756.60031 · doi:10.1016/0167-7152(92)90073-E
[25] Shao, Q. M., Complete convergence for \(\alpha \)-mixing sequences, Statistics & Probability Letters, 16, 4, 279-287 (1993) · Zbl 0787.60039 · doi:10.1016/0167-7152(93)90131-2
[26] Shao, Q. M., Maximal inequalities for partial sums of \(\rho \)-mixing sequences, Annals of Probability, 23, 2, 948-965 (1995) · Zbl 0831.60028 · doi:10.1214/aop/1176988297
[27] Szewczak, Z., On Marcinkiewicz-Zygmund laws, Journal of Mathematical Analysis and Applications, 375, 2, 738-744 (2011) · Zbl 1206.60034 · doi:10.1016/j.jmaa.2010.10.011
[28] Baek, J.-I.; Park, S.-T., Convergence of weighted sums for arrays of negatively dependent random variables and its applications, Journal of Theoretical Probability, 23, 2, 362-377 (2010) · Zbl 1196.60045 · doi:10.1007/s10959-008-0198-y
[29] Liang, H.-Y., Complete convergence for weighted sums of negatively associated random variables, Statistics & Probability Letters, 48, 4, 317-325 (2000) · Zbl 0960.60027 · doi:10.1016/S0167-7152(00)00002-X
[30] Liang, H.-Y.; Su, C., Complete convergence for weighted sums of NA sequences, Statistics & Probability Letters, 45, 1, 85-95 (1999) · Zbl 0967.60032 · doi:10.1016/S0167-7152(99)00046-2
[31] Kuczmaszewska, A., On complete convergence in Marcinkiewicz-Zygmund type SLLN for negatively associated random variables, Acta Mathematica Hungarica, 128, 1-2, 116-130 (2010) · Zbl 1224.60047 · doi:10.1007/s10474-009-9166-y
[32] Kruglov, V. M., Complete convergence for maximal sums of negatively associated random variables, Journal of Probability and Statistics, 2010 (2010) · Zbl 1206.60033 · doi:10.1155/2010/764043
[33] Ko, M.-H., On the complete convergence for negatively associated random fields, Taiwanese Journal of Mathematics, 15, 1, 171-179 (2011) · Zbl 1228.60041
[34] Chow, Y. S.; Teicher, H., Probability Theory: Independent, Interchangeability, Martingales (1997), New York, NY, USA: Springer, New York, NY, USA · Zbl 0891.60002 · doi:10.1007/978-1-4612-1950-7
[35] Baxter, J.; Jones, R.; Lin, M.; Olsen, J., SLLN for weighted independent identically distributed random variables, Journal of Theoretical Probability, 17, 1, 165-181 (2004) · Zbl 1050.60031 · doi:10.1023/B:JOTP.0000020480.84425.8d
[36] Pisier, G., Martingales with values in uniformly convex spaces, Israel Journal of Mathematics, 20, 3-4, 326-350 (1975) · Zbl 0344.46030 · doi:10.1007/BF02760337
[37] Huan, N. V.; Quang, N. V., The Doob inequality and strong law of large numbers for multidimensional arrays in general Banach spaces, Kybernetika, 48, 2, 254-267 (2012) · Zbl 1247.60038
[38] Assouad, P., Espaces \(p\)-lisses et \(q\)-convexes, inégalités de Burkholder, Proceedings of the Séminaire Maurey-Schwartz 1974-1975, Centre de Mathématiques Appliquées-École Polytechnique · Zbl 0318.46023
[39] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation, 27 (1987), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0617.26001
[40] Salem, R.; Zygmund, A., Some properties of trigonometric series whose terms have random signs, Acta Mathematica, 91, 245-301 (1954) · Zbl 0056.29001 · doi:10.1007/BF02393433
[41] Hill, J. D., The Borel property of summability methods, Pacific Journal of Mathematics, 1, 399-409 (1951) · Zbl 0043.28603 · doi:10.2140/pjm.1951.1.399
[42] Hanson, D. L.; Koopmans, L. H., On the convergence rate of the law of large numbers for linear combinations of independent random variables, Annals of Mathematical Statistics, 36, 559-564 (1965) · Zbl 0132.38604 · doi:10.1214/aoms/1177700167
[43] Pruitt, W. E., Summability of independent random variables, Journal of Applied Mathematics and Mechanics, 15, 769-776 (1966) · Zbl 0158.36403
[44] Franck, W. E.; Hanson, D. L., Some results giving rates of convergence in the law of large numbers for weighted sums of independent random variables, Bulletin of the American Mathematical Society, 72, 266-268 (1966) · Zbl 0141.16405 · doi:10.1090/S0002-9904-1966-11488-X
[45] Chow, Y. S., Some convergence theorems for independent random variables, Annals of Mathematical Statistics, 37, 1482-1493 (1966) · Zbl 0152.16905 · doi:10.1214/aoms/1177699140
[46] Chow, Y. S.; Lai, T. L., Limiting behavior of weighted sums of independent random variables, Annals of Probability, 1, 810-824 (1973) · Zbl 0303.60025
[47] Stout, W. F., Some results on the complete and almost sure convergence of linear combinations of independent random variables and martingale differences, Annals of Mathematical Statistics, 39, 1549-1562 (1968) · Zbl 0165.52702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.