Cramér's theorem (large deviations)

Template:Short description 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 $t\in ℝ$.

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

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

satisfies,

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

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

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 ${(ℒ(\frac{1}{n}\sum _{i=1}^n{X}_i))}_{n\in \mathbb{N}}$ satisfy a large deviation principle with rate function ${\mathrm{\Lambda }}^{\ast }$, where $ℒ(X)$ denotes the distribution of the random variable $X$.Template:Rp

References

• Template:Springer