A note on a maximal Bernstein inequality. (English) Zbl 1225.60032

Kevei, Péter; Mason, David M.

Bernoulli 17, No. 3, 1054-1062 (2011).

Summary: We show somewhat unexpectedly that whenever a general Bernstein-type maximal inequality holds for partial sums of a sequence of random variables, a maximal form of the inequality is also valid.

MSC:

60E15

Inequalities; stochastic orderings

Keywords:

Bernstein inequality; dependent sums; maximal inequality; mixing; partial sums

References:

[1]	Bentkus, R. and Rudzkis, R. (1980). On exponential estimates of the distribution of random variables. Lithuanian Math. J. 20 15-30 (in Russian). · Zbl 0428.60027
[2]	Deheuvels, P. and Mason, D.M. (1992). Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab. 20 1248-1287. · Zbl 0760.60028 · doi:10.1214/aop/1176989691
[3]	Deheuvels, P. and Mason, D.M. (1994). Functional laws of the iterated logarithm for local empirical processes indexed by sets. Ann. Probab. 22 1619-1661. · Zbl 0821.60042 · doi:10.1214/aop/1176988617
[4]	Doukhan, P. and Neumann, M.H. (2007). Probability and moment inequalities for sums of weakly dependent random variables, with applications. Stochastic Process. Appl. 117 878-903. · Zbl 1117.60018 · doi:10.1016/j.spa.2006.10.011
[5]	Einmahl, U. and Mason, D.M. (2000). An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13 1-37. · Zbl 0995.62042 · doi:10.1023/A:1007769924157
[6]	Einmahl, U. and Mason, D.M. (2005). Uniform in bandwidth consistency of kernel-type function estimators. Ann. Statist. 33 1380-1403. · Zbl 1079.62040 · doi:10.1214/009053605000000129
[7]	Giné, E. and Guillou, A. (2002). Rates of strong uniform consistency for multivariate kernel density estimators. En l’honneur de J. Bretagnolle, D. Dacunha-Castelle, I. Ibragimov. Ann. Inst. H. Poincaré Probab. Statist. 38 907-921. · Zbl 1011.62034 · doi:10.1016/S0246-0203(02)01128-7
[8]	Kallabis, R. and Neumann, M.H. (2006). An exponential inequality under weak dependence. Bernoulli 12 333-350. · Zbl 1126.62039 · doi:10.3150/bj/1145993977
[9]	Merlevède, F., Peligrad, M. and Rio, E. (2009). Bernstein inequality and moderate deviations under strong mixing conditions. In High Dimensional Probability V: The Luminy Volume (C. Houdré, V. Koltchinskii, D.M. Mason and M. Peligrad, eds.) 273-292. Beachwood, OH: IMS. · Zbl 1243.60019
[10]	Móricz, F.A. (1979). Exponential estimates for the maximum of partial sums. I. Sequences of rv’s. Special issue dedicated to George Alexits on the occasion of his 80th birthday. Acta Math. Acad. Sci. Hungar. 33 159-167. · Zbl 0394.60034 · doi:10.1007/BF01903391
[11]	Móricz, F.A., Serfling, R.J. and Stout, W.F. (1982). Moment and probability bounds with quasisuperadditive structure for the maximum partial sum. Ann. Probab. 10 1032-1040. · Zbl 0499.60052 · doi:10.1214/aop/1176993724
[12]	Rio, E. (2000). Théorie asymptotique des processus aléatoires faiblement dépendants. (French) Mathématiques & Applications (Berlin) 31 . Berlin: Springer. · Zbl 0944.60008
[13]	Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics . New York: Wiley. · Zbl 1170.62365
[14]	Saulis, L. and Statulevičius, V.A. (1991). Limit Theorems for Large Deviations . Dordrecht: Kluwer. · Zbl 0744.60028
[15]	Statulevičius, V.A. and Jakimavičius, D.A. (1988). Estimates for semiinvariants and centered moments of stochastic processes with mixing. I. Litovsk. Mat. Sb. 28 112-129; translation in Lithuanian Math. J. 28 67-80. · Zbl 0666.60027 · doi:10.1007/BF00972253
[16]	Stute, W. (1982). A law of the logarithm for kernel density estimators. Ann. Probab. 10 414-422. · Zbl 0493.62040 · doi:10.1214/aop/1176993866