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From the paper "What is a Random Sequence?" by Sergio B. Volchan, Math. Monthly 109, january 2002

Definition 3.1 An infinite binary sequence $x=x_1 x_2 \dots$ is random if it is collective; i.e., if it has the following two properties:

I. Let $f_n = \#\{m \leq n : x_m=1\}$ be the number of $1$s among the first $n$ terms in the sequence. Then $\lim_{n\to\infty}\frac{f_n}{n}=p$ exists and $0 < p < 1$.

II. If $\Phi : \{0, 1\}^*\to\{0,1\}$ is an admissible partial function (i.e., a rule for the selection of a subsequence of $x$ such that $x_n$ is chosen precisely when $\Phi(x_1 x_2 \dots x_{n-1}) = 1$), then the subsequence $x_{n_1} x_{n_2} \dots$ so obtained has property I for the same p.

[...] let $C(S,p)=\bigg\{ x \in \Sigma^\mathbb{N}:\forall\Phi\in S, \lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^{n}(\Phi x)_k=p \bigg\}$ where $0 < p < 1$, be the set of collectives with respect to $S$.

Theorem 3.2 (Wald) For any countable $S$ and any $p$ in $(0, 1)$, $\#C(S, p) = 2^{\aleph_0}$; that is, $C(S, p)$ has the cardinality of the continuum.

I would like to know more on this result. Is there any reference? Are there textbooks where to find it? How to prove it?

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    @anon yep, sorry. I've added the definition upon which the theorem is based, so there's no need to look at the article.2011-09-23

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Abraham Wald seems to have written two papers on collectives:

  • Sur la notion de collectif dans le calcul des probabilités, Comptes Rendus des Séances de l'Académie des Sciences, 202 (1936), pp. 1080-1083.

  • Die Wiederspruchsfreiheit des Kollektivbegriffes der Wahrscheinlichkeitsrechnung, Ergebnisse eines mathematischen Kolloquiums 8 (1937), pp. 38-72.

I’ve not seen either, but Volchan said that Wald proved that theorem in 1937, so it’s probably in the second of these papers.