Behaviors near explosion of nonlinear CSBPs with regularly varying mechanisms
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Bibliographic record
Abstract
We study the explosion phenomenon of nonlinear continuous-state branching processes (nonlinear CSBPs). First an explicit integral test for explosion is designed when the rate function does not increase too fast. We then exhibit three different regimes of explosion when the branching mechanism and the rate function are regularly varying respectively at $0$ and $\infty$ with indices $α$ and $β$ such that $0\leqα\leq β$ and $β>0$. If $α>0$ then the renormalisation of the process before its explosion is linear. When moreover $α\neq β$, the limiting distribution is that of a ratio of two independent random variables whose laws are identified. When $α=β$, the limiting random variable shrinks to a constant. Last, when $α=0$, i.e. the branching mechanism is slowly varying at $0$, the process is studied with the help of a nonlinear renormalisation. The limiting distribution becomes the inverse uniform distribution. This complements results known in the case of finite mean and provides new insight for the classical explosive continuous-state branching processes (for which $β=1$).
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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