0.1 Helly’s Selection Theorem
Helly’s Selection Theorem is the first of several results (including Tightness and its resulting theorems) forming a sort of compactness result for certain families of [[probability-measure|probability measures]]. The theorem essentially states that any sequence of distribution functions contains a subsequence that converges to a right-continuous non-decreasing function.
Any sequence of distribution functions \(\{ F_{n} \}\) contains a convergent subsequence \(\{ F_{n} \}\) contains a convergent subsequence \(\{ F_{n_{k}} \}: F_{n_{k}}(x) \to F(x)\) as \(k \to \infty\) for some right-continuous non-decreasing function \(F\). Convergence holds \(\forall x \in C(F)\).
0.1.0.1 Helly’s Selection Theorem & Weak Convergence
The limit \(F\) of a sequence of distribution function \(\{ F_{n} \}\) is itself a distribution function when the sequence \(\{ F_{n} \}\) is tight. This is helpful as we then have that any sequence of random variables \(\{ X_{n} \}\) whose distribution functions \(\{ F_{n} \}\) contain a subsequence \(\{ X_{n_{k}} \}\) who converge weakly.