Exponential tightness

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A sequence of probability measures $(\mu_n)_{n\ge1}$ on a topological space $E$ is exponentially tight at speed $(a_n)_{n\ge1}$ with $a_n\to\infty$ if for every $M>0$ there exists a compact set $K_M\subseteq E$ such that

\limsup_{n\to\infty}\frac{1}{a_n}\log \mu_n(K_M^{\,c}) \le -M.

Exponential tightness is a strengthened form of ordinary tightness for probability measures: it not only forces most mass into compacts, but does so with exponentially small tails at the LDP speed. It is frequently paired with a rate function to obtain or upgrade a large deviation principle, and it is a standard hypothesis in results like the Gärtner–Ellis theorem.

Examples

Let $X_1,X_2,\ldots$ be independent, identically distributed real random variables. If their moment generating function is finite on an open interval containing $0$ , then the laws of $\overline X_n=n^{-1}\sum_{i=1}^nX_i$ are exponentially tight at speed $n$ ; Chernoff bounds give exponential decay of both tails.
If $E$ itself is compact, then any sequence $(\mu_n)$ is exponentially tight at any speed, since one can take $K_M=E$ for all $M$ .