Chow-type maximal inequality for conditional demimartingales and its applications

Abstract

In this paper, the Chow-type maximal inequality for conditional demimartingales is established. By using the Chow-type maximal inequality, the authors provide the maximal inequality for conditional demimartingales based on concave Young functions. At last, the moment inequalities for conditional demimartingales are established.

Keywords

Conditional demimartingales / Chow-type maximal inequality / Concave Young functions

References

[1] Newman C. M., Wright A. L.. Associated random variables and martingale inequalities. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1982, 59(3): 361-371

[2] Christofides T. C.. Maximal inequalities for demimartingales and a strong law of large numbers. Statistics and Probability Letters, 2000, 50(4): 357-363

[3] Christofides T. C.. U-statistics on associated random variables. Journal of Statistical Planning and Inference, 2004, 119(1): 1-15

[4] Wang J. F.. Maximal inequalities for associated random variables and demimartingales. Statistics and Probability Letters, 2004, 66(3): 347-354

[5] Hu S. H., Wang X. J., Yang W. Z., Zhao T.. The Hàjek-Rènyi-type inequality for associated random variables. Statistics and Probability Letters, 2009, 79: 884-888

[6] Wang X. J., Hu S. H.. Maximal inequalities for demimartingales and their applications. Science in China Series A: Mathematics, 2009, 52(10): 2207-2217

[7] Wang, X. J., Hu, S. H., Zhao, T. and Yang, W. Z., Doob’s type inequality and strong law of large numbers for demimartingales, Journal of Inequalities and Applications, 2010, 2010, Article ID 838301, 11 pages.

[8] Journal of Inequalities in Pure and Applied Mathematics, 2007, 8 4

[9] Prakasa Rao B. L. S.. Remarks on maximal inequalities for non-negative demisubmartingales. Statistics and Probability Letters, 2012, 82(7): 1388-1390

[10] Prakasa Rao B. L. S.. Associated Sequences, Demimartingales and Nonparametric Inference, 2012, Basel AG: Springer- Verlag

[11] Christofides T. C.. Maximal inequalities for N-demimartingales. Archives of Inequalities and Applications, 2003, 50(1): 397-408

[12] Prakasa Rao B. L. S.. On some inequalities for N-demimartingales. Journal of the Indian Society of Agricultural Statistics, 2004, 57: 208-216

[13] Christofides T. C., Hadjikyriakou M.. Exponential inequalities for N-demimartingales and negatively associated random varibles. Statistics and Probability Letters, 2009, 79(19): 2060-2065

[14] Christofides T. C., Hadjikyriakou M.. Maximal and moment inequalities for demimartingales and N-demimartingales. Statistics and Probability Letters, 2012, 82(3): 683-691

[15] Hadjikyriakou M.. Marcinkiewicz-Zygmund inequality for nonnegative N-demimartingales and related results. Statistics and Probability Letters, 2011, 81(6): 678-684

[16] Wang X. J., Hu S. H., Prakasa Rao B. L. S., Yang W. Z.. Maximal inequalities for N-demimartingale and strong law of large numbers. Statistics and Probability Letters, 2011, 81(9): 1348-1353

[17] Hu S. H., Wang X. H., Yang W. Z., Wang X. J.. Some inequalities for demimartingales and Ndemimartingales. Statistics and Probability Letters, 2012, 82(2): 232-239

[18] Hadjikyriakou M.. Probability and Moment Inequalities for Demimartingales and Associated Random Variables. Dissertation, Department of Mathematics and Statistics, 2010, Nicosia: University of Cyprus

[19] Yuan D. M., Yang Y. K.. Conditional versions of limit theorems for conditionally associated random variables. Journal of Mathematical Analysis and Applications, 2011, 376(1): 282-293

[20] Roussas G. G.. On conditional independence, mixing, and association. Stochastic Analysis and Applications, 2008, 26(6): 1274-1309

[21] Prakasa Rao B. L. S.. Conditional independence, conditional mixing and conditional association. Annals of the Institute of Statistical Mathematics, 2009, 61(2): 441-460

[22] Agbeko N. K.. Concave function inequalities for sub-(super-)martingales. Ann. Univ. Sci. Budapest, Sect. Math, 1986, 29: 9-17