Access the full text.
Sign up today, get DeepDyve free for 14 days.
G. R. Shorack, J. A. Wellner (1986)
Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics
V. Bentkus, N. Kalosha, M. Zuijlen (2006)
On domination of tail probabilities of (super)martingales: Explicit boundsLithuanian Mathematical Journal, 46
(1997)
Fractional sums and integrals of r-concave tails and applications to comparison probability inequalities, Advances in stochastic inequalities (Atlanta, GA
V. Bentkus (2001)
An inequality for large deviation probabilities of sums of bounded i.i.d.r.v.Lithuanian Mathematical Journal, 41
Alan Jeffrey, G. Shorack, Jon Wellner (1987)
EMPIRICAL PROCESSES WITH APPLICATIONS TO STATISTICS (Wiley Series in Probability and Mathematical Statistics)Bulletin of The London Mathematical Society, 19
V. Paulauskas, A. Račkauskas (1989)
Approximation Theory in the Central Limit Theorem
M. Talagrand (1995)
The missing factor in Hoeffding's inequalitiesAnnales De L Institut Henri Poincare-probabilites Et Statistiques, 31
M. Ledoux (1999)
Concentration of measure and logarithmic Sobolev inequalities, 33
C. McDiarmid (1989)
Surveys in Combinatorics, 1989: On the method of bounded differences
V. Paulauskas, A. Rachkauskas (1989)
Approximation Theory in the Central Limit Theorem. Exact Results in Banach SpacesActa Applicandae Mathematica, 24
Ihrer Grenzgebiete, Theorie Der, Konvexen Körper (1975)
Ergebnisse der Mathematik und ihrer GrenzgebieteSums of Independent Random Variables
V. D. Milman, G. Schechtman (1986)
Lecture Notes in Mathematics
G. Shorack, J. Wellner (2009)
Empirical Processes with Applications to Statistics
V. Bentkus, M. Zuijlen (2003)
On Conservative Confidence IntervalsLithuanian Mathematical Journal, 43
I. Pinelis (1994)
Extremal Probabilistic Problems and Hotelling's $T^2$ Test Under a Symmetry ConditionAnnals of Statistics, 22
V. Bentkus (2003)
An Inequality for Tail Probabilities of Martingales with Differences Bounded from One SideJournal of Theoretical Probability, 16
V. Bentkus (2002)
A Remark on Bernstein, Prokhorov, Bennett, Hoeffding, and Talagrand InequalitiesLithuanian Mathematical Journal, 42
I. Pinelis (1998)
Optimal Tail Comparison Based on Comparison of Moments
M. Talagrand (1994)
Concentration of measure and isoperimetric inequalities in product spacesPublications Mathématiques de l'Institut des Hautes Études Scientifiques, 81
G. Pisier (1987)
ASYMPTOTIC THEORY OF FINITE DIMENSIONAL NORMED SPACES (Lecture Notes in Mathematics 1200)Bulletin of The London Mathematical Society, 19
V. Bentkus (1986)
Large deviations in Banach spacesProbability Theory Applications, 31
V. Bentkus (2001)
An Inequality for Large Deviation Probabilities of Sums of Bounded i.i.d. Random VariablesLithuanian Mathematical Journal, 41
M. Eaton (1970)
A Note on Symmetric Bernoulli Random VariablesAnnals of Mathematical Statistics, 41
V. Milman, G. Schechtman (1986)
Asymptotic Theory Of Finite Dimensional Normed Spaces
V. Bentkus (2002)
A remark on the inequalities of Bernstein, Prokhorov, Bennett, Hoeffding, and TalagrandLithuanian Mathematical Journal, 42
V. Bentkus (2002)
An Inequality for Tail Probabilities of Martingales with Bounded DifferencesLithuanian Mathematical Journal, 42
© Institute of Mathematical Statistics, 2004 ON HOEFFDING’S INEQUALITIES 1
V. Bentkus (2004)
On Hoeffding’s inequalitiesAnnals of Probabality, 32
S. Bobkov, F. Götze, C. Houdré (2001)
On Gaussian and Bernoulli covariance representationsBernoulli, 7
M. Eaton (1974)
A Probability Inequality for Linear Combinations of Bounded Random VariablesAnnals of Statistics, 2
I. Pinelis (1998)
High Dimensional Probability (Oberwolfach, 1996)
I. Pinelis (1999)
Advances in Stochastic Inequalities (Atlanta, GA, 1997)
I. Pinelis (2005)
On normal domination of (super)martingalesElectronic Journal of Probability, 11
V. Bentkus (1987)
Large Deviations in Banach SpacesTheory of Probability and Its Applications, 31
M. Ledoux (1997)
On Talagrand's deviation inequalities for product measuresEsaim: Probability and Statistics, 1
C. McDiarmid (1989)
Surveys in combinatorics, 1989 (Norwich, 1989)
N. Fisher, P. Sen (1994)
Probability Inequalities for Sums of Bounded Random Variables
V. Petrov (1975)
Sums of Independent Random Variables
M. Ledoux (1999)
Séminaire de Probabilités, XXXIII
Let M n = X 1 + ⋯ + X n be a martingale with bounded differences X m = M m − M m −1 such that ℙ{a m − σ m ≤ X m ≤ a m + σ m } = 1 with nonrandom nonnegative σ m and σ(X 1, …, X m −1)-measurable random variables a m . Write σ 2 = σ 1 2 + ⋯ + σ n 2 . Let I(x) = 1 − Φ(x), where Φ is the standard normal distribution function. We prove the inequalities $$\mathbb{P}\left\{ {M_n \geqslant x} \right\} \leqslant cI(x/\sigma ), \mathbb{P}\left\{ {M_n > x} \right\} \geqslant 1 - cI( - x/\sigma )$$ with a constant c such that 3.74 … ≤ c ≤ 7.83 …. The result yields sharp bounds in some models related to the measure concentration. In the case where all a m = 0 (or a m ≤ 0), the bounds for constants improve to 3.17 … ≤ c ≤ 4.003 …. The inequalities are new even for independent X 1, …, X n , as well as for linear combinations of independent Rademacher random variables.
Israel Journal of Mathematics – Springer Journals
Published: May 17, 2007
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.