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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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I misread this as "Looking for Waldo Theorem." That would have been fun. But anyway, if possible it might be nice to briefly explain what the theorem means instead of relying on a not-very-short article so the that question is more self-contained and provides better motivation. –  anon Sep 23 '11 at 8:45
    
I'm guessing it was in this one. –  J. M. Sep 23 '11 at 9:03
    
@anon yep, sorry. I've added the definition upon which the theorem is based, so there's no need to look at the article. –  Emanuele Natale Sep 23 '11 at 9:08
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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.

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