Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data
Bibliographic record
Abstract
Abstract The Marcinkiewicz strong law, lim n →∞ (1 / n 1/ p )∑ k =1 n ( D k - D ) = 0 almost surely with p ∈ (1, 2), is studied for outer products D k = { X k X ̅ k T }, where { X k } and { X ̅ k } are both two-sided (multivariate) linear processes (with coefficient matrices ( C l ), ( C ̅ l ) and independent and identically distributed zero-mean innovations {Ξ} and {Ξ̅}). Matrix sequences C l and C ̅ l can decay slowly enough (as | l | → ∞) that { X k , X ̅ k } have long-range dependence, while { D k } can have heavy tails. In particular, the heavy-tail and long-range-dependence phenomena for { D k } are handled simultaneously and a new decoupling property is proved that shows the convergence rate is determined by the worst of the heavy tails or the long-range dependence, but not the combination. The main result is applied to obtain a Marcinkiewicz strong law of large numbers for stochastic approximation, nonlinear function forms, and autocovariances.
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How this classification was reachedexpand
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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".