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Disjoint ideal fair-bit streams yield independent standard normals by Box–Muller

Statement

Assume the Axiom of Choice and Dependent Choice. On the countable product space of ideal independent fair bits, choose two disjoint infinite coordinate subsequences and form their bit-stream uniforms $U_{1}, U_{2}$ . Use the displayed formula when $0 < U_{1} < 1$ , and define $Z = (0, 0)$ when $U_{1} \notin (0, 1)$ , so the null endpoint event is assigned a value. Then $Z_{1} = \sqrt{- 2 \log U_{1}} \cos (2 π U_{2}), Z_{2} = \sqrt{- 2 \log U_{1}} \sin (2 π U_{2})$ are independent with the canonical standard-normal law $N (0, 1)$ .

Facts & Assumptions

Given: The Axiom of Choice, Dependent Choice, the fair-bit product construction, and the two Box–Muller formulas in the Statement.

[F1]

Disjoint fair-bit subsequences produce independent uniform variables under Countable Choice and Dependent Choice. Disjoint streams of independent fair bits produce independent uniform variables

[F2]

The Box–Muller transform of a joint-uniform pair has independent marginals with density $ϕ (x) = e^{- x^{2} / 2} / \sqrt{2 π}$ . Box–Muller has a product Gaussian density and Rayleigh second moment

[F3]

Under the Axiom of Choice, the canonical law $N (0, 1)$ is the probability measure with density $ϕ$ . Standard normal and normal laws

[F4]

A singleton has Lebesgue measure zero, so each uniform coordinate has zero probability of either endpoint. A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included

[A1]

The Axiom of Choice supplies Countable Choice and is used to identify the canonical $N (0, 1)$ law. The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )

[A2]

Dependent Choice is required by the countable fair-bit product. The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain

Proof

1.1givenF1F2F4A1A2

By [F1], the two disjoint subsequences give a joint-uniform pair on $[0, 1]^{2}$ ; [F4] shows the pair lies in $(0, 1)^{2}$ almost surely. The chosen endpoint convention changes no law. Applying [F2] gives a pair with independent marginals, each the probability law with density $ϕ$ . The existence of this finite bit pair uses Dependent Choice through [F1].

2.1F1F3step 1.1A1∎

By [A1], the Axiom of Choice supplies Countable Choice for the hypotheses of [F1] and is the exact axiom required by [F3]. Thus each density- $ϕ$ marginal is the canonical $N (0, 1)$ law, and the pair remains independent.

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Sources

Luc Devroye, Non-Uniform Random Variate Generation, Chapter V §4.4, generating normal random variates (standard reference, not scraped)
E. M. Raz, Lecture Notes in Probability, Chapter 5, Exercise 5.2 (standard reference, not scraped)