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:

$\mathrm{\Lambda }(t)=\log \mathrm{E}[\exp (t{X}_1)].$

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

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

${\mathrm{\Lambda }}^{*}(x):=\underset{t\in \mathbb{ℝ}}{\sup }(tx-\mathrm{\Lambda }(t))$

satisfies,

$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\log \left(P(\sum _{i=1}^n{X}_i\ge nx)\right)=-{\mathrm{\Lambda }}^{*}(x)$

for all $x>\mathrm{E}[X_1].$

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 ${({ℒ}(\frac{1}{n}\sum _{i=1}^n{X}_i))}_{n\in \mathbb{\mathbb{N}}}$ satisfy a large deviation principle with rate function ${\mathrm{\Lambda }}^{*}$.

References

• Klenke, Achim (2008). Probability Theory. Berlin: Springer. pp. 508. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_341. 
• Hazewinkel, Michiel, ed. (2001), "Cramér theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/c027000