Theorem

Cramér's Theorem (Large Deviations)

Probability and Statistics

Gives the exponential rate at which the probability that a sample average deviates far from its expected value decays, expressed through a rate function built from the moment generating function. Proved by Harald Cramer, it founded the theory of large deviations in probability.

Facts

Statement

For a sequence of independent and identically distributed real random variables with a finite moment generating function, the probability that the sample mean deviates from the true mean decays exponentially at a rate given by the theorem's rate function, the Legendre transform of the log moment generating function. 1

Proof Year

1938 1

Classification

Statement Form

Identity or Equation 1

Connections

Source Cramér's Theorem (Large Deviations) (Wikipedia)

Sources

1. Cramér's Theorem (Large Deviations) (Wikipedia)

Wikimedia Foundation

lead section, first paragraph
Cramér's theorem is a fundamental result in the theory of large deviations, a subdiscipline of probability theory. It determines the rate function of a series of iid random variables.
lead section, first paragraph, second sentence
A weak version of this result was first shown by Harald Cramér in 1938.
In Group: Probability Theory, Article text
Cramér's theorem is a fundamental result in the theory of large deviations, a subdiscipline of probability theory.
Proved By: Harald Cramér, Article text
A weak version of this result was first shown by Harald Cramér in 1938.

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Cramér's Theorem (Large Deviations)

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