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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All labels observed (2)

Label	Occurrences
Cramér–Chernoff method	2
Cramér’s theorem in large deviations canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T11560420 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Cramér’s theorem in large deviations
Context triple: [Harald Cramér, knownFor, Cramér’s theorem in large deviations]

A. Lyapunov central limit theorem
The Lyapunov central limit theorem is a version of the central limit theorem that provides sufficient moment conditions under which the normalized sum of independent (not necessarily identically distributed) random variables converges in distribution to a normal law.
B. Modern Probability Theory and Its Applications
"Modern Probability Theory and Its Applications" is a foundational textbook by Emanuel Parzen that systematically develops modern probability theory and demonstrates its use in a wide range of statistical and applied contexts.
C. Limit Laws for Sums of Independent Random Variables
Limit Laws for Sums of Independent Random Variables is a foundational mathematical work that systematically develops the theory of probability limit theorems, including results such as the law of large numbers and central limit behavior for sums of independent random variables.
D. Freidlin–Wentzell theory
Freidlin–Wentzell theory is a mathematical framework in probability that analyzes the behavior of stochastic dynamical systems under small random perturbations using large deviation principles.
E. Kolmogorov zero–one law
The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Cramér’s theorem in large deviations
Target entity description: 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.

A. Lyapunov central limit theorem
The Lyapunov central limit theorem is a version of the central limit theorem that provides sufficient moment conditions under which the normalized sum of independent (not necessarily identically distributed) random variables converges in distribution to a normal law.
B. Modern Probability Theory and Its Applications
"Modern Probability Theory and Its Applications" is a foundational textbook by Emanuel Parzen that systematically develops modern probability theory and demonstrates its use in a wide range of statistical and applied contexts.
C. Limit Laws for Sums of Independent Random Variables
Limit Laws for Sums of Independent Random Variables is a foundational mathematical work that systematically develops the theory of probability limit theorems, including results such as the law of large numbers and central limit behavior for sums of independent random variables.
D. Freidlin–Wentzell theory
Freidlin–Wentzell theory is a mathematical framework in probability that analyzes the behavior of stochastic dynamical systems under small random perturbations using large deviation principles.
E. Kolmogorov zero–one law
The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
F. None of above. chosen

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 ⓘ
isSpecialCaseOf	Gärtner–Ellis theorem ⓘ
namedAfter	Harald Cramér ⓘ
provides	good rate function for empirical mean ⓘ lower bound for large deviation probabilities ⓘ upper bound for large deviation probabilities ⓘ
relatedTo	Chernoff bounds ⓘ linked to: Chernoff bound Cramér–Chernoff method ⓘ linked to: Cramér’s theorem in large deviations Sanov’s theorem ⓘ central limit theorem ⓘ law of large numbers ⓘ
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 ⓘ queueing theory ⓘ risk theory ⓘ statistical mechanics ⓘ
usesConcept	Legendre–Fenchel transform ⓘ linked to: Legendre transformation convex analysis ⓘ logarithmic moment generating function ⓘ rate function ⓘ

How these facts were elicited

Referenced by (3)

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

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

Chernoff bound → relatedTo → Cramér–Chernoff method ⓘ

linked to: Cramér’s theorem in large deviations

Cramér’s theorem in large deviations → relatedTo → Cramér–Chernoff method ⓘ