Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 79 / 2

Work in a Hausdorff space $E$, for example a metric space. A rate function is a lower semicontinuous map $I:E\to [0,\infty ]$. A good rate function additionally has compact finite sublevel sets $\{x:I(x)\le r\}$.

A large deviation principle at large-deviation speed $L$ requires, for every open set $G$ and closed set $F$,The conventions are $\log 0=-\infty$ and $\inf \varnothing =+\infty$. A weak large deviation principle imposes the same lower bound but only the upper bound for compact sets. Exponential tightness supplies compact sets $K_M$ with $\mathrm{limsup}{L}^{-1}\log \mathbb{P}(X_L\notin K_M)<-M$ for every $M\ge 0$. The strict inequality can equivalently be obtained from the common non-strict definition by choosing a larger containment level.

Fix a finite $r\ge 0$ and choose $M>r$. By exponential tightness, choose a compact set $K_M$ with outside probability having upper exponential rate strictly below $-M$. A compact set in a Hausdorff space is closed, so its complement is an open set. The lower bound of the weak large deviation principle givesConsequently ${\inf }_{K_M^c}I>M$, and $\{I\le r\}\subset K_M$. Lower semicontinuity makes this sublevel set closed in $E$, hence a closed subset of a compact set. Thus it is compact. Every finite sublevel set is compact, so $I$ is a good rate function.

Let $F$ be any closed set. For each $M>0$, choose $K_M$ as above. The probability decompositionreduces the first term to a compact set, where the weak upper bound applies. For nonnegative numbers $u_L,v_L$, the inequality $u_L+v_L\le 2\max (u_L,v_L)$ shows that the upper exponential rate of their sum is no larger than the maximum of their upper exponential rates. HenceLet $M\to \infty$. This proves the full closed-set upper bound, including when ${\inf }_{F}I=+\infty$. The open-set lower bound was already assumed. Combining with part (b), the weak principle upgrades to a large deviation principle with good rate function $I$. This is exponential tightness upgrades a weak large deviation principle.