Bernstein inequalities

E354909

approximation theory result mathematical inequality probability inequality

Bernstein inequalities are fundamental results in approximation theory and probability that provide bounds on the derivatives or deviations of functions and random variables under certain smoothness or moment conditions.

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All labels observed (7)

Label	Occurrences
Bernstein inequalities canonical	1
Bernstein inequalities in approximation theory	1
Bernstein inequality for polynomials	1
Bernstein inequality on compact sets	1
Bernstein inequality on the circle	1
Bernstein inequality on the real line	1
Bernstein-type concentration inequality	1

Statements (49)

Predicate	Object
instanceOf	approximation theory result ⓘ mathematical inequality ⓘ probability inequality ⓘ
appliesTo	functions with bounded derivatives ⓘ independent random variables ⓘ polynomials on a bounded interval ⓘ
characterizedBy	dependence on variance and uniform bound of random variables ⓘ exponential decay of tail probabilities ⓘ trade-off between variance term and maximal bound term ⓘ
field	approximation theory ⓘ probability theory ⓘ
givesUpperBoundOn	supremum norm of derivatives of polynomials ⓘ tail probabilities of sums of random variables ⓘ
hasVariant	Bernstein inequalities self-linksurface differs ⓘ surface form: Bernstein inequality for polynomials Bernstein inequalities self-linksurface differs ⓘ surface form: Bernstein inequality on compact sets Bernstein inequalities self-linksurface differs ⓘ surface form: Bernstein inequality on the circle Bernstein inequalities self-linksurface differs ⓘ surface form: Bernstein inequality on the real line Bernstein inequalities self-linksurface differs ⓘ surface form: Bernstein-type concentration inequality classical Bernstein inequality for sums of independent random variables ⓘ matrix Bernstein inequality ⓘ
namedAfter	Sergei Natanovich Bernstein ⓘ
provides	bounds on derivatives of polynomials ⓘ bounds on deviations of sums of random variables ⓘ
relatedTo	Azuma–Hoeffding inequality ⓘ Bennett inequality ⓘ Bernstein polynomials ⓘ Chernoff bound ⓘ Chernoff bound ⓘ surface form: Hoeffding inequality Markov brothers' inequalities ⓘ
requiresCondition	bounded random variables or bounded moments ⓘ finite variance ⓘ smoothness conditions on functions ⓘ
typicalAssumption	bounded support or sub-exponential tails ⓘ independence of summands ⓘ zero-mean random variables ⓘ
usedFor	concentration of measure ⓘ confidence bounds for random sums ⓘ convergence rates of estimators ⓘ empirical process theory ⓘ error bounds in approximation theory ⓘ high-dimensional statistics ⓘ large deviations estimates ⓘ non-asymptotic probability bounds ⓘ statistical learning theory ⓘ uniform approximation of functions ⓘ
usedIn	analysis of randomized algorithms ⓘ machine learning generalization bounds ⓘ risk bounds for empirical risk minimization ⓘ signal processing and harmonic analysis ⓘ

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Felix Bernstein → notableWork → Bernstein inequalities ⓘ

Felix Bernstein → notableConcept → Bernstein inequalities ⓘ

this entity surface form: Bernstein inequalities in approximation theory