tight and relatively compact measures

Tight and relatively compact measuresFernando Sanz

Definition 1.

Let $\mathcal{M}=\{\mu_{i},i\in I\}$ be a family of finite measures on the Borel subsets of a metric space $\Omega$ . We say that $\mathcal{M}$ is tight iff for each $\epsilon>0$ there is a compact set $K$ such that $\mu_{i}(\Omega-K)<\epsilon$ for all $i$ . We say that $\mathcal{M}$ is relatively compact iff each sequence in $\mathcal{M}$ has a subsequence converging weakly to a finite measure on $\mathcal{B}(\Omega)$ .

If $\{F_{i},i\in I\}$ is a family of distribution functions, relative compactness or tightness of $\{F_{i}\}$ refers to relative compactness or tightness of the corresponding measures.

Theorem.

Let $\{F_{i},i\in I\}$ be a family of distribution functions with $F_{i}(\infty)-F_{i}(-\infty)<M<\infty$ for all $i$ . The family is tight iff it is relatively compact.

Proof.

Coming soon…(needs other theorems before) ∎

Title	tight and relatively compact measures
Canonical name	TightAndRelativelyCompactMeasures
Date of creation	2013-03-22 17:19:37
Last modified on	2013-03-22 17:19:37
Owner	fernsanz (8869)
Last modified by	fernsanz (8869)
Numerical id	6
Author	fernsanz (8869)
Entry type	Definition
Classification	msc 60F05
Related topic	LindebergsCentralLimitTheorem