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.Statement
The logarithmic moment generating function of a random variable is defined as:
Let be a sequence of iid real random variables with finite logarithmic moment generating function, e.g. for all.
Then the Legendre transform of :
satisfies,
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.
