Second phase changes in random $\boldsymbol{m}$-ary search trees and generalized quicksort: Convergence rates
Bibliographic record
Abstract
We study the convergence rate to normal limit law for the space requirement of random $m$-ary search trees. While it is known that the random variable is asymptotically normally distributed for $3\le m\le 26$ and that the limit law does not exist for $m>26$, we show that the convergence rate is $O(n^{-1/2})$ for $3\le m\le 19$ and is $O(n^{-3(3/2-\alpha)})$, where $4/3<\alpha<3/2$ is a parameter depending on $m$ for $20\le m\le 26$. Our approach is based on a refinement to the method of moments and applicable to other recursive random variables; we briefly mention the applications to quicksort proper and the generalized quicksort of Hennequin, where more phase changes are given. These results provide natural, concrete examples for which the Berry--Esseen bounds are not necessarily proportional to the reciprocal of the standard deviation. Local limit theorems are also derived.
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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 | 0.004 | 0.036 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".