Asymptotic Normality and Large Deviations

Abstract

Many sequences of random variables which turn up in statistics are asymptotically normal

$$ L\left( {{X_{n}}} \right)\sim N\left( {{x^{*}},\frac{1}{n} \cdot \varepsilon } \right). $$

In most practical cases it is worthwhile to look for more precise information about the asymptotic behaviour of the distributions. For the range of small deviations one can frequently use Edgeworth-expansions. For a description of the distributions in the range of large deviations one is led to a function K(x) which is usually called the entropy function. It turns out that in many cases one can go even further and find an asymptotic expansion similar to that one which has first been established by the so-called saddlepoint approximation (Daniels (1954)).