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. 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 .