{"id":"c343a292-360c-47b3-8f14-6ad24c3c8561","arxiv_id":"2506.01126","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Halfspace depth decay rates, for both the population and the empirical measure, encode whether the underlying distribution has light or heavy tails, and a plotting algorithm based on these rates can flag tail type.","lead":"This paper studies how a geometric measure of centralness, halfspace depth, decays as points move outward, and shows this decay reveals whether a distribution has light or heavy tails. It proves convergence rates for the sample version of this depth and turns them into a visual tool for classifying multivariate tail behavior.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 1's per-coordinate tail flags do not follow from the theorems: HD(t e_k, P) can be governed by a tilted halfspace involving other coordinates, so a heavy coordinate marginal can be flagged light. A concrete two-dimensional counterexample settles this.","rationale":"The reader's weakest_assumption identifies the same root issue: directional halfspace depth decay is controlled by the lightest feature among all halfspaces containing the point, not automatically by the corresponding coordinate marginal. Proposition 2.3 supplies only an upper bound, and the simulations in Section 4.2 already exhibit the failure mechanism: after an orthogonal rotation, all canonical marginals inherit the heavy t_3 component, yet the depth plots along e_2 and e_3 look exponential/Gaussian. The paper attributes this to coefficient magnitudes, but that is exactly the unproved assumption that the minimizing halfspace is coordinate-aligned or that no lighter tilted combination dominates. My counterexample makes the gap explicit: independent Pareto and near-cancelling Gaussian structure produce a halfspace containing t e_1 with Gaussian probability, so Algorithm 1 cannot distinguish a heavy coordinate marginal from a light linear combination. This does not call into question Theorem 3.1, whose empirical-to-population transfer appears plausible; it calls into question the methodology's central interpretative step. The other issues noted by the reader, such as strict inequalities in Theorem 2.5, the norm/supremum mismatch in Theorem 3.5, and the uncontrolled approximate depth in the illustrations, are real but secondary. They reinforce a conditional verdict without changing its direction. Since the main mathematical theorem is likely correct and the methodology gap is fixable by adding and verifying an explicit alignment or anti-cancellation condition, the conditional verdict should stand unchanged.","tokens_in":22894,"tokens_out":12420,"duration_ms":142594,"concrete_test":"Check the counterexample: let X_1 be standard Pareto(alpha) and X_2 = -X_1 + Z with Z ~ N(0,1), and compute, by numerical minimization over directions u with u_1 > 0, the population depth HD(t e_1, P), or simply use the upper bound HD(t e_1, P) <= P(X_1 + X_2 >= t) = P(Z >= t). If that bound is attained (or even only used), then log(1/HD(t e_1, P))/t -> infinity while log(1/P(X_1 >= t))/log t -> alpha, so Algorithm 1 flags e_1 as light despite a heavy coordinate marginal. A Monte Carlo version with large n and the same data-depth routine used in the paper should reproduce that flag; if it does not, the inferential gap is demonstrated computationally rather than only analytically.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Load-bearing concern: Section 4's Algorithm 1 reads the decay of HD(t_n e_k, P_n) as evidence about the k-th coordinate marginal, and the real-data conclusions in Section 4.3 (e.g. Prayagraj summer has the heaviest left tail) rely on that reading. Theorem 3.1 only shows the empirical ratio tracks the population halfspace depth; it says nothing about which halfspace achieves the infimum. Proposition 2.3 is an upper bound, not an equivalence; the minimizing halfspace for x = t e_k need not be the coordinate halfspace {y : y_k >= t}. It can be a tilted halfspace containing t e_k whose probability decays much faster than the k-th marginal. Concrete counterexample: in R^2 take X_1 Pareto with P(X_1 >= t) ~ t^{-alpha} and X_2 = -X_1 + Z with Z standard normal. The halfspace H = {y_1 + y_2 >= t} contains t e_1 and has P(H) = P(Z >= t) ~ (1/t) e^{-t^2/2}. Hence HD(t e_1, P) <= P(Z >= t), so (1/t) log(1/HD(t e_1, P)) >= t/2 - (log t)/t -> infinity, and Algorithm 1 flags e_1 as light-tailed even though X_1 is Pareto heavy. The same mechanism can affect any direction whenever a light linear combination has positive coefficient in that direction. The paper's own Section 4.2 discussion implicitly concedes this: although all marginals X_i are heavy (Y_3 ~ t_3 contributes to every canonical coordinate after rotation), the depth plots in e_2 and e_3 show exponential/Gaussian-looking decay, and the authors attribute this to coefficient sizes rather than to the marginal tail. Without an additional assumption, such as that the minimizing halfspace is the coordinate halfspace, or a lower bound excluding cancellation, the methodology does not establish per-coordinate tail discrimination. Theorem 3.1 itself is not affected; the failure is in the bridge from directional depth decay to coordinate-tail conclusions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the asymptotic decay of Tukey's halfspace depth for both the population and the empirical measure. It proves a general almost-sure uniform convergence result for the ratio of empirical to population halfspace depth at growing points (Theorem 3.1), derives population decay rates for light and heavy tails (Theorems 2.4, 2.5 and Proposition 2.6), and proposes a coordinate-wise algorithm (Algorithm 1) that flags directions as light- or heavy-tailed on the basis of the scaled log-depth plots. The methodology is illustrated on simulated rotated Gaussian/Laplace/Student data and on an outgoing longwave radiation dataset from two Indian cities across two seasons. The proofs rely on Alexander's weighted empirical process results and on the MRV framework of He and Einmahl (2017).","tokens_in":23184,"tokens_out":18087,"duration_ms":169383,"significance":"If Theorem 3.1 is correct, the paper provides a valuable complement to the heavy-tail depth results of He and Einmahl (2017) and the light-tail depth results of Burr and Fabrizio (2017), and the Alexander-based proof is elegant and reasonably general. The population bounds (Theorem 2.4, Theorem 2.5, Proposition 2.6) give useful comparison rates and the explicit example computations (Example 3.10) are instructive. However, the methodological contribution that motivates the title is not supported: Algorithm 1 interprets depth decay along a coordinate direction as a statement about that coordinate's marginal tail, which does not follow from the paper's theorems and is false in general. Since the real-data conclusions in Section 4.3 rest on this interpretation, the applied claims need substantial reworking.","major_comments":[{"comment":"The per-coordinate tail flags produced by Algorithm 1 are not consequences of the paper's theoretical results. Proposition 2.3 gives only an upper bound on HD(t e_k,P) in terms of the coordinate marginals; it does not identify the halfspace attaining (or nearly attaining) the infimum in (2). That halfspace may be tilted with respect to the coordinate axes. Concretely, in R^2 take X_1 Pareto with P(X_1 >= s) = s^{-alpha}, Z standard normal independent, and X_2 = -X_1 + Z. The halfspace H = {y_1 + y_2 >= t} contains t e_1 and has P(H) = P(Z >= t) ~ (1/t) e^{-t^2/2}; hence HD(t e_1,P) <= P(Z >= t), so (1/t) log(1/HD(t e_1,P)) -> infinity and Algorithm 1 flags e_1 as light-tailed, although the first coordinate marginal is Pareto heavy. This is not a pathological edge case: the simulation discussion in Section 4.2 explicitly observes that the depth in e_2 and e_3 appears exponential/Gaussian even though all marginals are heavy after rotation, and attributes the phenomenon to coefficient sizes. The real-data conclusions in Section 4.3 (e.g., 'Prayagraj summer has the heaviest left tail') rely on the same unjustified per-coordinate reading. The authors should either prove conditions under which the minimizing halfspace is aligned with the coordinate halfspace (e.g., for independent marginals with suitably ordered tails), or substantially weaken the interpretation of the algorithm and remove or qualify the climatological claims.","section":"Section 4.1, Algorithm 1; Section 4.3; Proposition 2.3"},{"comment":"The strict inequalities in (5) and (6) are stronger than what the proof establishes. The Markov-bound argument gives log(1/HD(tx,P)) >= -log C_N + t sup_{h in N} <x,h> (respectively >= -log M + t <x,h*>), so dividing by t and taking lim inf yields only '>='. Equality is possible if the upper bound is asymptotically tight, and the manuscript provides no additional argument ruling out equality. The statements should be weakened to non-strict inequalities, or the proof supplemented to exclude equality. The same issue propagates to Theorem 3.8 and equation (11).","section":"Theorem 2.5(i)-(ii) and proof in Section 5.1.3"},{"comment":"The proof of Theorem 3.5 displays the supremum over ||x|| = epsilon, while the statement requires a supremum over ||x|| = 1. Theorem 3.1 is formulated for a fixed epsilon-ball, so the proof needs to show explicitly that Condition (C2) can be verified on a set containing the unit sphere (for example by taking epsilon >= 1 and checking the condition uniformly on the unit ball) and that Proposition A.1 supplies the required uniform bound on that set. The current switch between epsilon and 1 leaves a gap that should be closed by a precise argument.","section":"Theorem 3.5 and proof in Section 5.2.2"},{"comment":"The proofs of Lemma 3.3 and Proposition 5.1 assume the existence of halfspaces attaining the infimum in the halfspace depth for an arbitrary probability measure P and for the empirical measure P_n. For a general probability measure the infimum over the family of halfspaces containing a given point need not be attained, because the map from the sphere to halfspace probabilities is only continuous under additional assumptions such as absolute continuity of P. The proofs should be reworked with minimizing sequences and an epsilon argument, or the result restricted to cases where attainment is guaranteed (e.g., absolutely continuous P and finite-support P_n).","section":"Lemma 3.3 and Proposition 5.1, proofs in Section 5.2.1"}],"minor_comments":[{"comment":"The word 'bevaiour' should be 'behaviour'.","section":"Section 4.3"},{"comment":"The word 'emcompasses' should be 'encompasses'.","section":"Remark 3.6"},{"comment":"The heading 'V apnik–˘Cervonenkis' contains a formatting artifact and should be 'Vapnik–Chervonenkis'.","section":"Appendix B"},{"comment":"The author 'Nagy' is cited inconsistently as 'Nagy, N. (2021)' in the Introduction and as 'Nagy, S.' in the reference list; the name should be normalized.","section":"References and Introduction"},{"comment":"The classification criteria in Algorithm 1 would benefit from a short explanation connecting the branches to Proposition 2.6, since the case y_n -> c > 0 corresponds to exponential decay, while the case y_n -> 0 requires the second-level w_n comparison to separate subexponential from polynomial behaviour.","section":"Algorithm 1 and Proposition 2.6"},{"comment":"The caption says 'QQ-plot of the marginals X_i' and the text states that the QQ-plot shows X_1, X_2, X_3 exhibit heavy-tailed behaviour; clarifying that the reference distribution is standard Laplace and that 'heavy-tailed' means deviation from Laplace would make the discussion easier to follow.","section":"Figure 2 caption and Section 4.2"}],"recommendation":"major_revision","confidential_remarks":"The theoretical core, especially Theorem 3.1 and the population decay bounds, appears sound and publishable once the proof gaps identified in the major comments are repaired. The most serious issue is the per-coordinate interpretation used in Section 4: the counterexample in Major Comment 1 shows that Algorithm 1 can flag a heavy-tailed marginal as light, and the real-data conclusions are therefore not supported. The authors should be encouraged either to add structural conditions that justify the coordinate-minimizer identification or to reposition the algorithm as a detector of light or heavy linear directions rather than of individual marginals, and to revise Section 4.3 accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the Singha–Kratz–Vadlamani paper. The headline: the central theorem is genuine and likely correct, but the tail-discrimination algorithm claims more than the theorems deliver.\n\nWhat's new: Theorem 3.1 unifies a.s. convergence rates for empirical halfspace depth at growing points under Alexander's weighted empirical processes, covering light and heavy tails in one framework. The proof is coherent, and the use of Alexander's machinery is appropriate. Theorem 2.5's population light-tail decay bounds are also new, and the paper is honest about what is borrowed from He–Einmahl and Burr–Fabrizio. The real-data illustration is a nice addition.\n\nSoft spots. First, Theorem 2.5(i)–(ii) state strict inequalities, but the proof only gives liminf ≥ sup; equality is not excluded. Restate with ≥ and the results are still useful. Second, Theorem 3.5's proof switches between sup over the unit sphere and an ε-ball without full verification of the uniform ball condition; fixable. Third, the simulations and real-data analysis use the Dyckerhoff approximate depth, and no tail-bound on the approximation error outside the data hull is given; that weakens the illustrations, not the theorems.\n\nThe load-bearing problem is the bridge to Algorithm 1. The paper treats the decay of HD(t e_k, P) as if it reveals the k-th marginal's tail. But Proposition 2.3 is an upper bound, not an equivalence, and Theorem 3.1 only transfers the empirical ratio to the population depth; it doesn't tell you which halfspace attains the infimum. The counterexample in the stress-test note is decisive: take X1 Pareto with tail index α and X2 = −X1 + Z with Z standard normal. Then the halfspace H = {y1 + y2 ≥ t} contains t e1 and has probability ~ P(Z ≥ t), so HD(t e1, P) decays like a Gaussian, and Algorithm 1 flags e1 as light-tailed even though X1 is heavy. The paper's own Section 4.2 discussion implicitly concedes the effect when it explains away non-Gaussian-looking depth decays by coefficient sizes. Without an added assumption—say, that the minimizing halfspace is the coordinate halfspace, or a lower bound that rules out cancellation—the per-coordinate tail flags, including the OLR conclusion, are not supported.\n\nThe citation pattern is fine; self-citation is not an issue here. This deserves a serious referee. I'd send it to an expert in depth functions and empirical processes, with a request to check the Theorem 3.1 proof detail and to require either a proof of the per-coordinate bridge or a clear heuristic caveat on Algorithm 1.","headline":"A genuinely useful unification of empirical halfspace depth rates via Alexander's weighted empirical processes, but the per-coordinate tail-discrimination algorithm claims more than the theorems prove.","tokens_in":23872,"tokens_out":3159,"would_cite":true,"duration_ms":30165,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F15","62G20","62G32","62H05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the empirical (sample) halfspace depth tracks the population halfspace depth at growing points almost surely, making the depth-decay rate a reliable signal of light versus heavy multivariate tails.","keywords":["halfspace depth","Tukey depth","tail behaviour","multivariate regular variation","empirical processes","weighted empirical processes","extreme quantiles","depth contours"],"falsifier":"Take the paper's rotated example: independent Gaussian, Laplace, and $t_3$ marginals rotated so every coordinate $X_i$ has a nonzero $t_3$ coefficient, making all three marginal distributions heavy-tailed. Proposition 2.3 and Theorem 2.5 imply that along $e_2$, $\\mathrm{HD}(t e_2,P)$ decays at least exponentially because the Gaussian direction has nonzero inner product with $e_2$. Plotting $\\frac{1}{t}\\log(1/\\mathrm{HD}(t e_2,P_n))$ on a large sample should show the light-tail signature even though the $e_2$ marginal is heavy; observing that would settle that directional flags are not per-coordinate marginal tail statements.","tokens_in":2055,"feed_emoji":"📉","tokens_out":2051,"duration_ms":110183,"temperature":0.7,"pith_summary":"The paper's goal is to use halfspace depth, the smallest probability mass in any half-space containing a point, as a probe for how a multivariate distribution's tails behave. The authors establish that for growing points $t_n x$, the sample depth $\\mathrm{HD}(t_n x, P_n)$ converges almost surely to the population depth $\\mathrm{HD}(t_n x, P)$, under conditions on a threshold sequence $\\gamma_n$, with no assumption on the tail type. This transfers the population decay rate to the empirical depth: exponential-type decay signals a light tail, and polynomial decay $t^{-\\alpha}$ signals a heavy tail under multivariate regular variation. On this basis they propose an exploratory algorithm that plots $\\frac{1}{t_n}\\log(1/\\mathrm{HD}(t_n e_k, P_n))$ along coordinate directions to flag directions as light- or heavy-tailed, and they illustrate the method on simulated and real climate data.","feed_headline":"Sample depths reveal whether tails are light or heavy","feed_subtitle":"The empirical halfspace depth at growing points matches the population depth almost surely, so its decay rate reads the tail.","key_machinery":"The central object is halfspace depth, $\\mathrm{HD}(x,P)=\\inf_{H\\ni x}P(H)$, equivalently written as the minimum over unit vectors $h$ of the projected tail probabilities $1-F_h(\\langle h,x\\rangle)$ and $F_h(\\langle h,x\\rangle)$. The argument is carried by a weighted empirical process bound for ratio empirical processes indexed by halfspaces: Lemma 3.3 controls the depth ratio by $\\sup_{H\\in\\mathcal{H}_{t_n x}}|P_n(H)/P(H)-1|$, and condition (C2) restricts that supremum to halfspaces with $P(H)\\ge\\gamma_n$, where the convergence is uniform by a theorem on weighted empirical processes indexed by a VC class of sets. The capacity function $g_c$ of that theorem enters only through condition (C1a), and the authors show that it plays a negligible role compared with the tail-driven choice of $\\gamma_n$.","core_discovery":"The anchor result, Theorem 3.1, says that if the depth at the growing points stays above a threshold $\\gamma_n$ that is not too small compared with the sample size, then $\\sup_{\\|x\\|\\le\\varepsilon}\\big|\\mathrm{HD}(t_n x, P_n)/\\mathrm{HD}(t_n x, P)-1\\big|\\to 0$ almost surely. Combined with the population results of Section 2, this yields explicit rates: under multivariate regular variation with index $\\alpha$, $\\mathrm{HD}(t_n x, P_n)/(1-P(t_n B^d))$ converges almost surely to the limiting depth $\\mathrm{HD}(x,\\nu)$, giving polynomial decay; if the moment generating function is finite in some direction, the sample depth obeys an exponential-type lower bound. The comparison of these rates gives a way to tell light from heavy tails from a single sample, within the data hull and without extrapolation.","pith_inferences":["The per-coordinate reading of directional flags is the fragile part of the method: because Proposition 2.3 bounds the depth along any direction by the lightest marginal contributing to that direction, a rotated heavy marginal can be masked by a lighter component, as the paper's own rotated simulation illustrates. A more robust tool would scan many random projections and report the lightest decay f","The heavy-tail case could be pushed from qualitative discrimination to quantitative estimation: in the multivariate regular variation regime, regressing $\\log\\mathrm{HD}(t_n x,P_n)$ on $\\log t_n$ over a range where the theorem's conditions hold would yield an estimator of the tail index $\\alpha$, a step the paper leaves implicit.","The choice of $\\gamma_n$ and hence the usable range of $t_n$ is governed by conditions (C1a)-(C2); this suggests a principled stopping rule for the plots, based on when the sample depth falls below the computable threshold, instead of the visual inspection currently used.","Because halfspace depth does not characterize the full joint distribution, the tool's verdict concerns which halfspace probabilities decay fastest; pairing it with a test for multivariate regular variation, as the authors themselves caution, is needed before interpreting a 'heavy' flag as a full tail statement."],"forward_implications":["For heavy-tailed multivariate regularly varying distributions, the empirical halfspace depth satisfies $\\mathrm{HD}(t_n x, P_n)/(1-P(t_n B^d))\\to\\mathrm{HD}(x,\\nu)$ almost surely, so the sample depth inherits the $t_n^{-\\alpha}$ decay of the population depth.","For light-tailed distributions with a finite moment generating function in some direction, $\\liminf_n \\frac{1}{t_n}\\log(1/\\mathrm{HD}(t_n x, P_n))>\\langle x,h^*\\rangle$ almost surely, so the empirical depth cannot decay faster than exponential in that direction.","In the Gaussian example, choosing $\\gamma_n=n^{-\\beta}$ with $0<\\beta<1$ gives $t_n\\le\\sqrt{2\\beta\\log n}$, and both the lower and the marginals-based upper bounds are of order $n^{-\\beta}$, showing the transfer from population to sample rate is sharp up to constants.","The ratio bound of Lemma 3.3, together with the weighted empirical process theorem, transfers population rates to sample rates without requiring Dvoretzky-Kiefer-Wolfowitz inequalities or tail-specific assumptions.","The proposed algorithm flags a direction as light-tailed when $\\frac{1}{t_n}\\log(1/\\mathrm{HD}(t_n e_k,P_n))$ grows or stabilizes at a positive constant, and heavy-tailed when the log-log version stabilizes at a finite positive slope; if every coordinate is heavy, the conclusion is that the distribution is heavy-tailed."],"supporting_citations":[{"why":"Supplies the weighted empirical process theorem (Theorem 5.1) and capacity function that yield the almost sure ratio bound under conditions (C1a)-(C1b).","marker":"Alexander (1987)"},{"why":"Provides the MRV population halfspace depth limit recalled as Proposition A.1, which the paper extends to the empirical depth under multivariate regular variation.","marker":"He and Einmahl (2017)"},{"why":"Establishes the MRV framework, the capacity bound $g_c(t)=O(1)$, and the extrapolation beyond the data hull that the paper compares with its within-hull approach.","marker":"Einmahl et al. (2015)"},{"why":"Gives the projection form of halfspace depth used in identity (1) and the almost sure convergence of sample depth to population depth.","marker":"Donoho and Gasko (1992)"},{"why":"Records the projection property that underlies Proposition 2.3 and supplies the approximate depth computation used in the examples.","marker":"Dyckerhoff (2004)"},{"why":"Provides the light-tail uniform convergence rates for halfspace depth that serve as the benchmark for the light-tail results.","marker":"Burr and Fabrizio (2017)"},{"why":"Shows that halfspace depth does not characterize the underlying distribution, motivating the use of empirical measures, for which Struyf and Rousseeuw (1999) give a characterization.","marker":"Nagy (2021)"}],"fun_headline_variants":["Halfspace depth rates reveal light or heavy tails","Empirical halfspace depth reads tail decay rates","Tail weight seen through halfspace depth convergence","Sample depth decay distinguishes tail heaviness","Halfspace depth: how to tell light from heavy tails"],"cache_read_input_tokens":25728,"weakest_assumption_plain":"The algorithm reads each coordinate direction's depth decay as that coordinate's own tail statement, but the upper bound in Proposition 2.3 lets the lightest marginal contributing to a direction control the decay, so a heavy marginal can be masked by a lighter one in the same rotated direction.","fun_headline_variants_meta":{"raw":{"variants":["Halfspace depth rates reveal light or heavy tails","Empirical halfspace depth reads tail decay rates","Tail weight seen through halfspace depth convergence","Sample depth decay distinguishes tail heaviness","Halfspace depth: how to tell light from heavy tails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000537,"raw_usage":{"total_tokens":2546,"prompt_tokens":882,"completion_tokens":1664,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":1594}},"tokens_in":498,"tokens_out":1664,"duration_ms":12610,"temperature":1.0,"reasoning_tokens":1594,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:53:38.720928+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the paper's rotated example: independent Gaussian, Laplace, and $t_3$ marginals rotated so every coordinate $X_i$ has a nonzero $t_3$ coefficient, making all three marginal distributions heavy-tailed. Proposition 2.3 and Theorem 2.5 imply that along $e_2$, $\\mathrm{HD}(t e_2,P)$ decays at least exponentially because the Gaussian direction has nonzero inner product with $e_2$. Plotting $\\frac{1}{t}\\log(1/\\mathrm{HD}(t e_2,P_n))$ on a large sample should show the light-tail signature even though the $e_2$ marginal is heavy; observing that would settle that directional flags are not per-coordinate marginal tail statements.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the weighted empirical process theorem (Theorem 5.1) and capacity function that yield the almost sure ratio bound under conditions (C1a)-(C1b)."},{"cited_title":"Li, and R","cited_arxiv_id":null,"evidence_quote":"Establishes the MRV framework, the capacity bound $g_c(t)=O(1)$, and the extrapolation beyond the data hull that the paper compares with its within-hull approach."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the projection form of halfspace depth used in identity (1) and the almost sure convergence of sample depth to population depth."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Records the projection property that underlies Proposition 2.3 and supplies the approximate depth computation used in the examples."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the light-tail uniform convergence rates for halfspace depth that serve as the benchmark for the light-tail results."}],"review_version":1}