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Manuscript Psy04-B@1 · Psy04-B-v1.tex · theorem · 62c2b6f0

Theorem: Anytime-valid version

Finite-sample exclusion conditional on the cut · lines 2204–2231 · proof included or linked

Psy04-B@1#thm:anytime

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Suppose the conditional null bounds hold indefinitely. Set

r_{N}^{any} = \frac{lo g ( N ( N + 1 ) / δ )}{2 N} .

Then

P_{null} {\exists N \geq 1 : \frac{1}{N} i = 1 \sum^{N} (S_{i} - c_{i}) > r_{N}^{any}} \leq δ .

Stopping at a first crossing is therefore valid.

Proof. At $N$ , allocate $δ_{N} = δ / [N (N + 1)]$ . Apply Theorem thm:test, then use

N \geq 1 \sum \frac{1}{N ( N + 1 )} = 1

and a union bound.

Displayed formulas in this passage

unlabeled · line 2208
$r_{N}^{any} = \frac{lo g ( N ( N + 1 ) / δ )}{2 N} .$
unlabeled · line 2214
$P_{null} {\exists N \geq 1 : \frac{1}{N} i = 1 \sum^{N} (S_{i} - c_{i}) > r_{N}^{any}} \leq δ .$
unlabeled · line 2227
$N \geq 1 \sum \frac{1}{N ( N + 1 )} = 1$

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