Bibliographic record
Abstract
1. It is tempting to believe that any sequence (?n) that is C6saro-convergent in probability necessarily has a subsequence that is a.s. C6saro-convergent. This is not true however. As an example, take an independent identically distributed sequence that satisfies the weak but not the strong law of large numbers (e.g., any symmetric distribution without first moment but with tails slightly smaller than Cauchy). 2. If we drop the nonnegativity, Observation 1 becomes false. Consider, for example, the constants (1)n log n. But if every permutation of a sequence of arbitrarily signed random variables is a.s. C6saro-convergent to a finite limit, does the sequence satisfy condition (A)? We do not know. 3. S. D. Chatterji [4] has already given versions of the subsequence theorem for ?n in LP with p < 1, but with a factor n-'/P instead of n-1. A nice review paper is [5]. 4. More recently, E. P6ter [7] gave sufficient criteria describing general distributional limit laws for which a permutation invariant version of the subsequence principle holds, in the same way that Berkes's result improves Koml6s's theorem.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.015 | 0.002 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".