Cramér's theorem (large deviations)

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 is defined as: Let

be a sequence of iid real random variables with finite logarithmic moment generating function, i.e.

Then the Legendre transform of

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