A Berry–Esseen bound for vector-valued martingales
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Bibliographic record
Abstract
This note provides a conditional Berry–Esseen bound for the sum of a martingale difference sequence {Xi}i=1n in Rd, d≥1, adapted to a filtration {Fi}i=1n. We approximate the conditional distribution of S=∑i=1nXi given a sub-σ-field F0⊂F1 by that of a mean zero normal random vector having the same conditional variance given F0 as the vector S. Assuming that the conditional variances E[XiXi⊤∣Fi−1], i≥1, are F0-measurable and non-singular, and the third conditional moments of ‖Xi‖, i≥1, given F0 are uniformly bounded, we present a simple bound on the conditional Kolmogorov distance between S and its approximation given F0 which is of order Oa.s.([ln(ed)]5/4n−1/4).
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it