# zbMATH — the first resource for mathematics

A general class of exponential inequalities for martingales and ratios. (English) Zbl 0942.60004
The author derives several new exponential inequalities for a martingale difference sequence $$(d_i,F_i)$$, which satisfies either $$E[|d_j|^k\mid F_{j-1}]\leq (k!/2)\sigma^2_j c^{k-2}$$ or $$P(|d_j|\leq c\mid F_{j-1})= 1$$, for $$k>2$$, $$0< c<\infty$$, where $$\sigma^2_j= E[d^2_j\mid F_{j-1}]$$. For example, put $$V^2_n= \sum^n_{j=1} \sigma^2_j$$ and $$M_m= \sum^n_{j=1} d_j$$, and then for all $$x,y>0$$, $P(M_n\geq x, V^2_n\text{ for some }n)\leq \exp\Biggl\{-{x^2\over 2(y+ cx)}\Biggr\}.$ The proof of the above and related results is based on decoupling techniques introduced by S. Kwapien and W. A. Woyczynski [in: Almost everywhere convergence, 237-265 (1989; Zbl 0693.60033)] and further developed by the author and others. The author gives also results for continuous time square integrable martingales. He also shows that in the special case of conditionally symmetric random variables the integrability conditions on the sequence $$d_j$$ can be relaxed.

##### MSC:
 60E15 Inequalities; stochastic orderings 60G42 Martingales with discrete parameter 60G44 Martingales with continuous parameter
Full Text:
##### References:
 [1] BARLOW, M. T., JACKA, S. D. and YOR, M. 1986. Inequalities for a pair of processes stopped at a () random time. Proc. London. Math. Soc. 3 52 142 172. · Zbl 0585.60055 [2] BENNETT, G. 1962. Probability inequalities for sums of independent random variables, J. Amer. Statist. Assoc. 57 33 45. · Zbl 0104.11905 [3] BURKHOLDER, D. L. 1991. Exploration in martingale theory and its applications. Ecole d’Ete de Ṕrobabilites de Saint-Flour XIX. Lecture Notes in Math. 1464 1 66. Springer, Berlin. \' [4] CABALLERO, M. E., FERNANDEZ, B. and NUALART, D. 1996. Estimation of densities and applicaťions. Univ. Barcelona Math. Preprint Ser. 222. [5] CHOW, Y. S. and TEICHER, H. 1988. Probability Theory: Independence, Interchangeability, Martingales, 2nd ed. Springer, New York. · Zbl 0652.60001 [6] DE LA PENA, H. 1994. A bound on the moment generating function of a sum of dependent variables with an application to simple random sampling without replacement. Ann. Inst. H. Poincare Probab. Statist. 30 197 211. \' · Zbl 0796.60020 [7] DE LA PENA, V. H. 1995. A bound on the moment generating function of a sum of dependent Z variables with an application to simple random sampling without replacement. Cor. rection. Ann. Inst. H. Poincare. Probab. Statist. 31 703 704. \' · Zbl 0830.60012 [8] DE LA PENA, V. H. 1996a. A new class of exponential inequalities I. [9] DE LA PENA, V. H. 1996b. A new class of exponential inequalities II. [10] FREEDMAN, D. 1975. On tail probabilities for martingales. Ann. Probab. 3 100 118. · Zbl 0313.60037 [11] HITCZENKO, P. 1990a. Upper bounds for the L -norms of martingales. Probab. Theory Related p Fields 86 225 238. · Zbl 0677.60017 [12] HITCZENKO, P. 1990b. Best constants in martingale version of Rosenthal’s inequality. Ann. Probab. 18 1656 1668. · Zbl 0725.60018 [13] HOEFFDING, W. 1963. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13 30. JSTOR: · Zbl 0127.10602 [14] KHOSHNEVISAN, D. 1996. Deviation inequalities for continuous martingales. Stochastic Process. Appl. · Zbl 0889.60043 [15] KWAPIEN, S. and WOYCZYNSKI, W. A. 1989. Tangent sequences of random variables: basic \' ínequalities and their applications. In Proceedings of Conference on Almost EveryZ where Convergence in Probability and Ergodic Theory G. A. Edgar and L. Sucheston,. eds. 237 265. Academic Press, New York. · Zbl 0693.60033 [16] KWAPIEN, S. and WOYCZYNSKI, W. A. 1992. Random Series and Stochastic Integrals: Single and \' Ḿultiple. Birkhauser, Boston. \" · Zbl 0751.60035 [17] LEVENTHAL, S. A. 1989. A uniform CLT for uniformly bounded families of martingale differences. J. Theoret. Probab. 2 271 287. · Zbl 0681.60023 [18] MCKEAN, H. P. 1962. A Holder condition for Brownian local time. J. Math. Kyoto Univ. 1 \" 195 201. · Zbl 0121.13101 [19] PINELIS, I. 1992. An approach to inequalities for the distributions of infinite-dimensional Z martingales. In Probability in Banach Spaces 8 R. M. Dudley, M. G. Hahn and. J. Kuelbs, eds. 128 134. Birkhauser, Boston. \" · Zbl 0793.60016 [20] PINELIS, I. 1994. Optimum bounds for the distributions of martingales in Banach spaces. Ann. Probab. 22 1679 1706. · Zbl 0836.60015 [21] PINELIS, I. 1995. Sharp exponential inequalities for the martingales in the 2-smooth Banach spaces and applications to “scalarizing” decoupling. In Probability in Banach Spaces 955 70 J. Hoffmann-Jorgensen, J. Kuelbs and M. Marcus, eds. Birkhauser, Boston. \" Z. [22] PINELIS, I. and UTEV, S. A. 1989. Exact exponential bounds for sums of independent random variables. Theory Probab. Appl. 34 304 346. · Zbl 0693.60036 [23] REVUZ, D. and YOR, M. 1991. Continuous Martingales and Brownian Motion. Springer, Berlin. · Zbl 0731.60002 [24] SHIRYAYEV, A. N. 1984. Probability. Springer, Berlin. · Zbl 0536.60001 [25] WANG, G. 1989. Some sharp inequalities for conditionally symmetric martingales. Ph.D. dissertation. Univ. Illinois, Urbana-Champaign. [26] WISE, G. L. and HALL, E. B. 1993. Counterexamples in Probability and Real Analysis. Oxford Univ. Press, New York. · Zbl 0827.26001 [27] NEW YORK, NEW YORK 10027 E-MAIL: vp@wald.stat.columbia.edu
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.