Bibliographic record
Abstract
In this thesis, we investigate the automatic continuity properties of law-invariant risk measures on general model spaces. In Chapter 2, we study automatic order lower semi-continuity of law-invariant risk measures, which is usually termed as the Fatou property in the literature. Let X be a rearrangement-invariant space other than L∞ over a non-atomic probability space. We show that every real-valued, law-invariant, coherent risk measure automatically has the Fatou property at every random variable X ∈ X whose negative tails have vanishing norm (i.e., limn∥X1{X≤−n}∥ = 0) if and only if X satisfies the Almost Order Continuous Equidistributional Average (AOCEA) property, namely, d(CL(X), Xa) = 0 for any nonnegative random variable X ∈ X , where CL(X) is the convex hull of all random variables having the same distribution as X and Xa = { X ∈ X : limn∥X1|X|≥n∥ = 0 } . We also show that the AOCEA property is satisfied by most classical model spaces, including Orlicz spaces. In Chapter 3, we first show that on an r.i. space with the AOCEA property, every real-valued, law-invariant, coherent risk measure is automatically σ(X , X ′ )-lower semicontinuity at every random variable X ∈ X whose negative tails have vanishing norm. Here X ′ is the associated space of X . We also recover a local version of the Fenchel-Moreau Duality and apply it to establish automatic dual representations of risk measures. Finally, in Chapter 4, we apply our results to study when law-invariant bounded linear functionals automatically collapse to the mean, i.e., being scalar multiples of the expectation. We show that on every r.i. space with the AOCEA property, a bounded law-invariant linear functional collapses to the mean. We also construct an r.i. space on which a bounded law-invariant linear functional may fail to collapse to the mean and thus the space fails the AOCEA property. The thesis is based on [10, 11].
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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 | 0.003 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.002 |
| 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".