Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 26 / 2

A large deviation principle for random elements $X^L$ in a Hausdorff space $E$, at large-deviation speed $a_L\to \infty$, has a lower semicontinuous rate function $I:E\to [0,\infty ]$. For every open set $G$ and every closed set $F$,Equivalently, for every Borel set $A$, the lower and upper bounds use its interior and closure, respectively. The conventions are $\inf \varnothing =+\infty$ and $\log 0=-\infty$. A good rate function has compact sublevel sets $\{I\le r\}$ for every finite $r$. Here take $a_L=L$; the different speeds in the moderate deviation principle are stated explicitly when used.

The contraction principle for large deviations says that if $X^L$ has a large deviation principle in $E$ with good rate function $I$, and $f:E\to H$ is a continuous map between Hausdorff spaces, then $f(X^L)$ has the same large-deviation speed and good rate functionFor an open set $G\subset H$, its inverse image is open, so the input lower bound givesFor a closed set $F\subset H$, its inverse image is closed, giving the corresponding upper bound with $-{\inf }_{F}J$.

It remains to prove goodness, including the endpoint issue in the infimum. Each fibre $f^{-1}(\{y\})$ is closed. If $J(y)<\infty$, compact sublevel sets and lower semicontinuity make $I$ attain its infimum on that fibre: intersect it with the compact set $\{I\le J(y)+1\}$ and use the closed nested sublevels tending down to $J(y)$. Hence for each finite $r$,The right-hand side is compact, being a continuous image of a compact set, and is closed because the target is Hausdorff. Thus $J$ is lower semicontinuous and good, completing the proof.

The family is exponentially tight at large-deviation speed $a_L$ if for every $M>0$ there is a compact set $K_M\subset E$ such thatFor this question $a_L=L$. The definition says that escape from an appropriate compact set can be made negligible at any prescribed exponential rate. Requiring a strict inequality instead gives the same notion, by choosing a larger $M$.

Use the standard Hausdorff state-space convention, so compact sets are closed. Let $F$ be any closed set and choose $K_M$ from exponential tightness. The weak large deviation principle applies to the compact set $F\cap K_M$, whileThe principle of the largest exponential term givesLet $M\to \infty$. This is the full closed-set upper bound; the open-set lower bound was already present.

For goodness, fix $r<\infty$ and choose $M>r$. The complement $K_M^c$ is open. Its lower bound and its exponential-tightness upper estimate implyThus ${\inf }_{K_M^c}I\ge M$ and $\{I\le r\}\subset K_M$. Lower semicontinuity makes this sublevel set closed, so it is a closed subset of a compact set and therefore compact. This proves that exponential tightness upgrades a weak large deviation principle to the full principle with the same good rate function.