Large deviations theory

Remarks

Here $\log 0 = -\infty$. A sequence of random elements $Z_n$ of $\mathcal{X}$ satisfies the principle when their laws do. The factor $1/n$ is the speed; some applications use another speed $1/\varepsilon$ or $1/a_n$ with $a_n \to \infty$. If $A$ is a Borel set whose interior $A^\circ$ and closure $\bar A$ satisfy $I(A^\circ) = I(\bar A)$, then $\frac{1}{n} \log \mu_n(A) \to -I(A)$.

Example: let $\mu_n$ be the normal law with mean $0$ and variance $1/n$ on $\R$, and let $I(x) = x^2/2$. For an interval $[a, \infty)$ with $a > 0$, the Gaussian tail estimate gives $\frac{1}{n} \log \mu_n([a, \infty)) \to -a^2/2 = -I([a, \infty))$. The same tail estimate gives both bounds for all closed and open sets, so $(\mu_n)$ satisfies the principle with this $I$.

Non-example: the same sequence does not satisfy the principle with $I(x) = x^2$. The upper bound fails for the closed set $F = [1, \infty)$. Indeed $\frac{1}{n} \log \mu_n(F) \to -1/2$, while $-I(F) = -1 < -1/2$.