information:filtering, information:classification, linguistic:voice_change
Subsequences

Theorem — Let random variables X, X,... ∈ L (Ω) be such that X → 0 weakly in L (Ω) and X → 1 weakly in L (Ω). Then there exist integers n < n < ⋯ such that X n 1 + ⋯ + X n k k {\displaystyle {\frac {X_{n_{1}}+\cdots +X_{n_{k}}}{\sqrt {k}}}} converges in distribution to N (0, 1) {\textstyle {\mathcal {N}}(0,1)} as k tends to infinity.