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

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 $M$.