Cramér's theorem (large deviations) explained

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 (which is the cumulant-generating function) of a random variable is defined as:

Λ (t) =log \operatorname E [\exp(tX _{1)].}

Let

X _{1,} X _{2,} ... be a sequence of iid real random variables with finite logarithmic moment generating function, i.e. Λ (t) < inf ty for all t \in R .

Then the Legendre transform of

Λ :Λ ^{*(x):=} \sup _{t} \left(tx -Λ (t) \right)

satisfies,

\lim _{n} (1 n)/(log) \left(P\left(\sum ^{n} _{i =1} X _{i} \geq nx \right)\right) = -Λ ^{*(x)}

for all

x > \operatorname E[X _{1].}

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

If

X _{1,} X _{2,} ... is a series of iid random variables, then the distributions \left( l L ( \tfrac 1 n \sum ^{n} _{i =1} X _{i)} \right) _{n} satisfy a large deviation principle with rate function Λ ^{*} .

References

• Book: Klenke . Achim . 2008 . Probability Theory . limited . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6. 508 .