Previously Unrecognized Links Between Statistical Dependence and Some Common Modes of Convergence
Why this work is in the frame
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Bibliographic record
Abstract
Most introductory courses in probability include the definitions of, and relationships between, convergence in distribution, probability, and mean square. Surprisingly, there are also links between the apparently unrelated notion of statistical dependence and two of these modes of convergence. These links are virtually absent from both the research literature and probability texts. Our goal here is to fill this gap, and by so doing provide some insight into the three modes of convergence. In particular, we establish ultimate dependence between members of a sequence of convergent random variables and its nondegenerate limit when convergence is in probability and in mean square. Through simple examples and graphical displays, we convey the heuristics that underlie the main results.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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