On the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Unbounded Random Variables
Bibliographic record
Abstract
For a sequence of i.i.d. unbounded random variables {Y n , n ≥ 1} and a constant b > 1, it is shown for that if 𝔼(log (max {|Y 1|, e})) < ∞, then and for almost every ω ∈ Ω, where l and L are the essential infimum of Y 1 and the essential supremum of Y 1, respectively. For the case where 𝔼(log (max {|Y 1|, e})) = ∞, examples are given wherein the limit point set ℂ is identified and it is not necessarily the interval [l, L]. The current work is a follow-up to the investigation of Li, Qi, and Rosalsky (Stochastic Analysis and Applications, 2008, 28:86–102) identifying the limit point set of W n when Y 1 is bounded; the results for unbounded Y 1 are structurally different from those for bounded Y 1 and are thus not merely simple extensions of the bounded case.
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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 | 0.005 | 0.021 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.005 | 0.001 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".