zbMATH — the first resource for mathematics

\(L_p\)-version of the Dubins-Savage inequality and some exponential inequalities. (English) Zbl 1166.60016
Summary: The Dubins-Savage inequality is generalized by using the \(p\)th \((1<p\leq 2)\) conditional moment of the martingale differences. This inequality is further extended under suitable conditions when \(p>2\). Another martingale inequality due to Freedman is also generalized when \(0<p\leq 2\). Implications of these inequalities for strong convergence are discussed. Some general exponential inequalities are also given for martingales (supermartingales) under suitable conditions.

60E15 Inequalities; stochastic orderings
60G42 Martingales with discrete parameter
60F15 Strong limit theorems
PDF BibTeX Cite
Full Text: DOI
[1] Chatterji, S.D.: An L p -convergence theorem. Ann. Math. Stat. 40, 1068–1070 (1969) · Zbl 0176.48101
[2] De la Peña, V.H.: A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27, 537–564 (1999) · Zbl 0942.60004
[3] Doob, J.L.: An inequality useful in martingale theory. Sankhya, Ser. A 35, 1–4 (1973) · Zbl 0272.60039
[4] Dubins, L.E., Savage, L.J.: A Tchebycheff-like inequality for stochastic processes. Proc. Nat. Acad. Sci. USA 53, 274–275 (1965) · Zbl 0131.35104
[5] Freedman, D.: A remark on the strong law. Ann. Probab. 2, 324–327 (1974) · Zbl 0321.60025
[6] Freedman, D.: On tail probabilities for martingales. Ann. Probab. 3, 100–118 (1975) · Zbl 0313.60037
[7] Kallenberg, O.: On the existence and path properties of stochastic integrals. Ann. Probab. 3, 262–280 (1975) · Zbl 0307.60050
[8] Khan, R.A.: Some remarks on Blackwell–Ross martingale inequalities. Probab. Eng. Inf. Sci. 21, 109–115 (2007) · Zbl 1112.60033
[9] Khan, R.A., Tomkins, J.: Refinements of the Dubins–Savage inequality. J. Theor. Probab. 13, 659–672 (2000) · Zbl 0966.60008
[10] Neveu, J.: Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco (1965) · Zbl 0137.11301
[11] Neveu, J.: Discrete Parameter Martingales. North-Holland, Amsterdam (1975) · Zbl 0345.60026
[12] Stout, W.F.: Almost Sure Convergence. Academic Press, San Diego (1974) · Zbl 0321.60022
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.