Cramér’s theorem in large deviations

E933485

large deviations principle mathematical theorem result in probability theory

Cramér’s theorem in large deviations is a fundamental result in probability theory that characterizes the exponential decay rate of tail probabilities for sums of independent, identically distributed random variables via a convex rate function.

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Statements (47)

Predicate	Object
instanceOf	large deviations principle ⓘ mathematical theorem ⓘ result in probability theory ⓘ
appliesTo	independent identically distributed random variables ⓘ partial sums of random variables ⓘ
assumes	existence of moment generating function in a neighborhood of zero ⓘ identical distribution of summands ⓘ independence of summands ⓘ
characterizes	exponential decay of tail probabilities ⓘ large deviation probabilities ⓘ
concerns	empirical mean of i.i.d. random variables ⓘ probabilities of deviations from the mean ⓘ
describes	asymptotic behavior of probabilities of rare events ⓘ logarithmic asymptotics of tail probabilities ⓘ
field	large deviations theory ⓘ probability theory ⓘ
formalizes	exponential tightness of sums of i.i.d. variables ⓘ
framework	large deviation principle on the real line ⓘ
generalizationOf	classical exponential tail bounds for sums of i.i.d. variables ⓘ
hasProperty	rate function has compact level sets ⓘ rate function is convex ⓘ rate function is lower semicontinuous ⓘ
historicalPeriod	20th century ⓘ
implies	law of large numbers type behavior at exponential scale ⓘ
inspired	development of modern large deviations theory ⓘ
introducedBy	Harald Cramér NERFINISHED ⓘ
isSpecialCaseOf	Gärtner–Ellis theorem NERFINISHED ⓘ
namedAfter	Harald Cramér NERFINISHED ⓘ
provides	good rate function for empirical mean ⓘ lower bound for large deviation probabilities ⓘ upper bound for large deviation probabilities ⓘ
relatedTo	Chernoff bounds NERFINISHED ⓘ Cramér–Chernoff method NERFINISHED ⓘ Sanov’s theorem NERFINISHED ⓘ central limit theorem NERFINISHED ⓘ law of large numbers NERFINISHED ⓘ
states	empirical mean satisfies a large deviation principle with a convex good rate function ⓘ
typicalFormulation	logarithmic asymptotics for probabilities of empirical mean in Borel sets ⓘ
usedIn	finance ⓘ information theory NERFINISHED ⓘ queueing theory ⓘ risk theory ⓘ statistical mechanics ⓘ
usesConcept	Legendre–Fenchel transform NERFINISHED ⓘ convex analysis ⓘ logarithmic moment generating function ⓘ rate function ⓘ

Referenced by (1)

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Harald Cramér → knownFor → Cramér’s theorem in large deviations ⓘ