Solution
= 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 that
$$
\boxed{\limsup_L\frac1{a_L}\log\mathbb P(X^L\notin K_M)\le-M.}
$$
For 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 $M$.