Cramér's theorem (large deviations)

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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. A weak version of this result was first shown by Harald Cramér in 1938.


The logarithmic moment generating function (which is the cumulant-generating function) of a random variable as:

Let be a series of iid real random variables with finite logarithmic moment generating function, e.g. for all .

Then the Legendre transform of :


for all

In the terminology of the theory of large deviations the result can be reformulated as follows:

If is a series of iid random variables, then the distributions satisfy a large deviation principle with rate function .