Contraction principle

Contraction principle: Let $(\mu_n)$ be probability measures on a space $X$ that satisfy a large deviation principle with speed $a_n\to\infty$ and rate function $I\colon X\to[0,\infty]$. Let $f\colon X\to Y$ be continuous, and let $\nu_n=\mu_n\circ f^{-1}$ be the pushforward measures on $Y$. Then $(\nu_n)$ satisfies an LDP on $Y$ with the same speed $a_n$ and rate function

with the convention $\inf\varnothing=+\infty$.

In terms of random variables, if $Z_n$ satisfies an LDP on $X$ and $Y_n=f(Z_n)$ with $f$ continuous, then $(Y_n)$ satisfies an LDP with rate obtained by minimizing $I$ over the fiber $\{x:f(x)=y\}$. This principle is routinely combined with Sanov's theorem and Cramér's theorem to derive LDPs for many statistics.