◀ Prev · Parts · Full · Next ▶

Theorem (Abraham Wald, 1936, 1937)cite-ref-3[3] If there are only countably many admissible rules, then almost any sequence is a collective.

Proof sketch: Use measure-theoretic probability.

Fix one admissible rule. Sample a random sequence from Bernoulli space. With probability 1 (use martingales), the subsequence picked by the admissible rule still has lim n 1 n ∑ ∑ i = 1 n x m i = p {\displaystyle \lim _{n}{\frac {1}{n}}\sum _{i=1}^{n}x_{m_{i}}=p} . Now add all the countably many rules. With probability 1, each subsequence picked by each rule still has lim n 1 n ∑ ∑ i = 1 n x m i = p {\displaystyle \lim _{n}{\frac {1}{n}}\sum _{i=1}^{n}x_{m_{i}}=p} .

However, this definition was found not to be strong enough. Intuitively, the long-time average of a random sequence should oscillate on both sides of p {\displaystyle p} , like how a random walk should cross the origin infinitely many times. However, Jean Ville showed that, even with countably many rules, there exists a binary sequence that tends towards p {\displaystyle p} fraction of ones, but, for every finite prefix, the fraction of ones is less than p {\displaystyle p} .cite-ref-4[4]

Ville's Construction (Jean Ville, 1939) There exists a collective with countably many admissible rules such that, for all n {\displaystyle n} , 1 n ∑ ∑ k = 1 n x k ≤ ≤ p {\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}x_{k}\leq p} . cite-ref-5[5]

◀ Prev · Parts · Full · Next ▶