zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A note on a maximal Bernstein inequality. (English) Zbl 1225.60032
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:
60E15Inequalities in probability theory; stochastic orderings
WorldCat.org
Full Text: DOI arXiv
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 · numdam:AIHPB_2002__38_6_907_0 · eudml:77748
[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 · euclid: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.
[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