On Cantrell–Rosalsky’s strong laws of large numbers
Bibliographic record
Abstract
Starting from a result of almost sure (a.s.) convergence for nonnegative random variables, a simplified proof of a strong law of large numbers (SLLN) established in ([Cantrell, A., Rosalsky, A., 2003. Some strong law of large numbers for Banach space valued summands irrespective of their joint distributions. Stochastic Anal. Appl. 21, 79–95], Theorem 1) for random elements in a real separable Banach space is presented, and some other results of a.s. convergence related to SLLNs in ([Cantrell, A., Rosalsky, A., 2004. A strong law for compactly uniformly integrable sequences of independent random elements in Banach spaces. Bull. Inst. Math. Acad. Sinica 32, 15–33], Th. 3.1) and ([Cantrell, A., Rosalsky, A., 2003. Some strong law of large numbers for Banach space valued summands irrespective of their joint distributions. Stochastic Anal. Appl. 21, 79–95], Theorem 2) are derived. No conditions of independence or on the joint distribution of random elements are required. Likewise, no geometric condition on the Banach space where random elements take values is imposed. Some applications to weighted (for an array of constants) sums of random elements and to the case of random sets are also considered.
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How this classification was reachedexpand
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 | 0.003 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".