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 (which is the cumulant-generating function) of a random variable is defined as:
: \Lambda(t)=\log \operatorname E exp(tX_1)

Let $X_1, X_2, \dots$ be a sequence of iid real random variables with finite logarithmic moment generating function, i.e. $\Lambda(t) < \infty$ for all $t \in \mathbb R$.

Then the Legendre transform of $\Lambda$:
:$\Lambda^*(x):= \sup_ \left(tx-\Lambda(t) \right)$

satisfies,

:$\lim_ \frac 1n \log \left(P\left(\sum_^n X_i \geq nx \right)\right) = -\Lambda^*(x)$

for all x > \operatorname E _1

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

If $X_1, X_2, \dots$ is a series of iid random variables, then the distributions $\left(\mathcal L ( \tfrac 1n \sum_^n X_i) \right)_$ satisfy a large deviation principle with rate function $\Lambda^*$.

References

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*{{springer, title=Cramér theorem, id=p/c027000

Large deviations theory
Theorems in probability theory