Exponential tightness

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

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:

• If $X$ has a finite moment generating function in a neighborhood of $0$, then the laws of $\bar X_n=\frac1n\sum_{i=1}^n X_i$ are typically exponentially tight at speed $n$ on $\mathbb R$ (exponential moment bounds imply exponentially small 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 $M$.