= Exponential tightness
= Exponentially tight
{synonym}
A family with <large-deviation speed> $a_N$ is exponentially tight if for every $M>0$ there is a <compact set> $K_M$ such that
$$
\limsup_N a_N^{-1}\log\mathbb P(X_N\notin K_M)\leq-M.
$$
Local upper bounds then imply the closed-set upper bound of a <large deviation principle>: intersect the closed set with $K_M$, cover that compact intersection by finitely many neighborhoods on which the local bounds hold, and use a finite union bound. The remaining probability has exponent at most $-M$; let $M\to\infty$.
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