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 $\Lambda$:

$\Lambda ^{*}(x):=\sup _{t\in \mathbb {R} }\left(tx-\Lambda (t)\right)$

satisfies,

$\lim _{n\to \infty }{\frac {1}{n}}\log \left(P\left(\sum _{i=1}^{n}X_{i}\geq nx\right)\right)=-\Lambda ^{*}(x)$

for all $x>\operatorname {E} [X_{1}].$

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

If $X_{1},X_{2},\dots$ is a series of iid random variables, then the distributions $\left({\mathcal {L}}({\tfrac {1}{n}}\sum _{i=1}^{n}X_{i})\right)_{n\in \mathbb {N} }$ satisfy a large deviation principle with rate function $\Lambda ^{*}$, where ${\mathcal {L}}(X)$ denotes the distribution of the random variable $X$.

References

• "Cramér theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]