Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 31 / 1

Put $s_i={\sup }_{C_i}f$ and omit empty sets. Since the sets cover the space, boundedness givesFor nonnegative numbers $u_{n,i}$, their sum lies between their maximum and m times their maximum. Consequently its scaled logarithmic upper limit is at most ${\max }_i{\mathrm{limsup}}_n{n}^{-1}\log u_{n,i}$; the added $n^{-1}\log m$ vanishes. Each $C_i$ is closed, so the upper bound in the large deviation principle givesThe convention $\log 0=-\infty$ also handles sets of zero probability or infinite infimum rate.

Take $m=\lceil (b-a)/\epsilon \rceil$. The ceiling, as printed in the PDF, makes the intervals cover the bounded range of f. If $a=b$, use the single set $\mathcal{X}$ instead, since f is constant. By continuity their preimages are closed. On every nonempty preimage $C_i$, the oscillation of f is at most epsilon. Hence for every $y\in C_i$,Taking the supremum over y in the second expression gives the bound for ${\sup }_{C_i}f-{\inf }_{C_i}I$; if that infimum is infinite, the bound is automatic. Apply the finite-cover estimate to obtainThis is the exponential-integral upper estimate in the bounded form of Varadhan lemma.

The continuity of f makes its prescribed interval preimage B open, and $\hat{x}\in B$. On B we have $f(x)>f(\hat{x})-\epsilon$, soThe open-set lower bound of the large deviation principle therefore yieldsThus the claimed lower bound holds at every chosen point. When $I(\hat{x})=\infty$ its right side is minus infinity and the assertion is immediate.

Let epsilon decrease to zero in the upper estimate. In the lower estimate first take the supremum over all chosen points and then decrease epsilon to zero. The bounds coincide, proving Varadhan lemma:The value is finite: f is bounded, $I\ge 0$, and the large-deviation bounds on the whole space force $\inf I=0$. This proof actually uses bounded continuity and the large deviation principle; compact sublevels are not additionally needed for this bounded version.