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
StatementFor 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 Connections
Sources
1. Cramér's Theorem (Large Deviations) (Wikipedia)
Wikimedia Foundationlead 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.
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