On the Strong Convergence and Complete Convergence for Pairwise NQD Random Variables
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Bibliographic record
Abstract
Let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mfenced open="{" close="}" separators="|"><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:math>be a sequence of positive constants with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo>↑</mml:mo></mml:math>and let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mfenced open="{" close="}" separators="|"><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:math>be a sequence of pairwise negatively quadrant dependent random variables. The complete convergence for pairwise negatively quadrant dependent random variables is studied under mild condition. In addition, the strong laws of large numbers for identically distributed pairwise negatively quadrant dependent random variables are established, which are equivalent to the mild condition<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:msubsup><mml:mo stretchy="false">∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mi>P</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mfenced open="|" close="|" separators="|"><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:mfenced><mml:mo>></mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mrow><mml:mo><</mml:mo><mml:mi>∞</mml:mi></mml:math>. Our results obtained in the paper generalize the corresponding ones for pairwise independent and identically distributed random variables.
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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 | 0.006 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 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 it