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.

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

${\displaystyle \Lambda (t)=\log \operatorname {E} [\exp(tX_{1})].}$

Let ${\displaystyle X_{1},X_{2},\dots }$ be a sequence of iid real random variables with finite logarithmic moment generating function, i.e. ${\displaystyle \Lambda (t)<\infty }$ for all ${\displaystyle t\in \mathbb {R} }$.

Then the Legendre transform of ${\displaystyle \Lambda }$:

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

satisfies,

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

for all ${\displaystyle x>\operatorname {E} [X_{1}].}$^[1]: 508

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

If ${\displaystyle X_{1},X_{2},\dots }$ is a series of iid random variables, then the distributions ${\displaystyle \left({\mathcal {L}}({\tfrac {1}{n}}\sum _{i=1}^{n}X_{i})\right)_{n\in \mathbb {N} }}$ satisfy a large deviation principle with rate function ${\displaystyle \Lambda ^{*}}$, where ${\displaystyle {\mathcal {L}}(X)}$ denotes the distribution of the random variable ${\displaystyle X}$.^[1]: 514

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