Moderate Deviations

Moderate Deviations

Consider the familiar simple set up for the central limit theorem (CLT, see Central Limit Theorems). Let ${X}_{1},{X}_{2},\ldots$ be independently and identically distributed real random variables with common distribution function F(x). Let ${Y }_{n}\,=\, \frac{1} {n}({X}_{1} + \cdots + {X}_{n}),n\,=\,1,2,\ldots$. Suppose that

$$ \int \nolimits \nolimits xF(dx) = 0,\int \nolimits \nolimits {x}^{2}F(dx) = l $$

(1)

Then the central limit theorem states that

$$P\left (\vert {Y }_{n}\vert > \frac{a} {\sqrt{n}}\right ) \rightarrow 2[1 - \Phi (a)]$$

(2)

where $\Phi (x) = \frac{1} {\sqrt{2\pi }}{ \int \nolimits \nolimits }_{-\infty }^{x}\exp (-{t}^{2}/2)dt$ and a > 0.

In other words, the CLT gives an approximation to the two-sided deviation of size $\frac{a} {\sqrt{n}}$ of Y _n and the approximation is a number in (1 / 2, 1). Deviations of the this type are called ordinary deviations.

However, one needs to study deviations larger than ordinary deviations to...