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

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.