Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)
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A canonical random walk from product increments

Example

Assume countable choice and dependent choice. On the product space Ω={0,1}N, give each coordinate the fair Bernoulli law and use the product probability measure of Assuming countable and dependent choice, countable products of arbitrary probability spaces. Write Xn(x)=xn, set ξn=2Xn−1∈{−1,1}, and put S0=0 and Sn=∑r=0n−1ξr for n≥1. The process (Sn)n≥0 is the canonical simple random walk built from independent increments.

Verification

Given: Countable choice, dependent choice, and the fair Bernoulli coordinate laws.

[F1] The countable product theorem supplies the product probability measure with its prescribed finite marginals (Assuming countable and dependent choice, countable products of arbitrary probability spaces).

[F2] Its coordinate maps are independent (Coordinate random elements of a countable product are independent).

1.1F1F2given

The countable product theorem gives the probability measure with fair Bernoulli coordinate marginals. The coordinate-independence corollary makes the Xn independent. Thus the signs ξn are independent, with P(ξn=1)=P(ξn=−1)=1/2.

2.1step 1.1∎

For any finite list of distinct increment times, their joint law is the uniform law on the corresponding sign vectors. The process coordinates are the partial sums of these increments, not the increments themselves.

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