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) 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 TightAndRelativelyCompactMeasures 2013-03-22 17:19:37 2013-03-22 17:19:37 fernsanz (8869) fernsanz (8869) 6 fernsanz (8869) Definition msc 60F05 LindebergsCentralLimitTheorem