Papers

For any fixed $k$, we prove a lower bound on the $k$th largest modulus of an eigenvalue of the non-backtracking matrix $B$. Specifically, consider any deterministic or random family of graphs that converges locally to the unimodular Galton-Watson tree with root degree distribution $D$, and set $\kappa:=\mathbb E[D(D-1)]/\mathbb E[D]$. Given $\kappa>1$ and an exponential-moment bound on the empirical degree distributions, we show that $|\lambda_k(B)|\geq\sqrt{\kappa}-o_N(1)$, where $N$ is the number of vertices. When restricted to locally tree-like regular graphs, this recovers a well-known consequence of the Ihara-Bass formula. In the specific case where the graph is generated through the Erdős-Rényi model with expected degree $d>1$, this proves a conjecture of Bordenave, Lelarge, and Massoulié. To do this, we show that the normalized log-determinant of the Bethe-Hessian of the graph is bounded by that of the Bethe-Hessian of its local limit. This bound is violated if the eigenvalues of the non-backtracking matrix are too small. We establish this using an effective-conductance interpretation of the tree Green's function recursion.

Large language model (LLM) distillation aims to transfer the capabilities of a powerful teacher to a smaller student. Direct imitation, however, can also transfer the teacher's systematic bias and errors. This challenge is particularly pronounced under covariate shift, when the teacher's reliability on target questions is uncertain and target-domain reward feedback is unavailable. We propose Coupled Calibration and Learning (CCL), an LLM distillation algorithm that couples teacher calibration with student updates through token-level branching, using reward feedback only on source questions. Each iteration calibrates the teacher using source feedback and then uses the calibrated teacher to train the student on target questions. The updated student, in turn, informs subsequent calibration. In an autoregressive policy framework, we prove that the output student's expected average Kullback-Leibler divergence to the oracle student converges to zero at a polynomial rate in the number of iterations. The oracle maximizes the true reference-regularized target reward within the student class, which need not represent the unrestricted optimal policy. Our analysis quantifies the progress of projected student gradient updates while controlling the error in teacher calibration. We further establish a separation from regularized direct matching: its error relative to the oracle student can remain bounded away from zero even when the teacher achieves higher regularized target reward than every student policy. These results demonstrate that LLM distillation can overcome persistent teacher bias and recover the optimal student through coupled calibration and learning, without target-domain reward feedback.

Covariates missing not at random generally prevent point identification of regression coefficients without untestable restrictions. This paper studies linear regression when every missing covariate is restricted to a prespecified compact interval. The resulting population target is the set of best linear predictor coefficients compatible with the observed-data law. Under a non-atomic observed-data law and the stated regularity conditions, we can represent this identified set as the image, under the least-squares moment map, of the Aumann expectation of a random moment set. For the population set and its empirical analogue, we derive explicit bounds under bounded, sub-exponential and polynomial envelope conditions. The bounds show their dependence on sample size, dimension, confidence level and tail parameters. Under a Donsker condition and a uniform zero-set error bound, an oracle enlargement of the set-valued Z-estimator also converges at the root n rate. We separately study a Hadamard-design method for exploring the effect of admissible imputations without enumerating all vertices and build intuition to detect situations where the population target can be approximated by a zonotope. A numerical experiment illustrates how co-missingness and the width of the imputation intervals affect this diagnostic.

By consuming multiple copies of an unknown qubit state, one can modify its purity while preserving the direction of its Bloch vector. We determine the maximum linear rate at which qubit states of different purities can be interconverted, allowing a nonzero error, quantified, for instance, by the trace distance, provided that it vanishes in the limit of infinitely many copies. Interestingly, the optimal conversion rate is determined by the two eigenvalues of the complex right-logarithmic-derivative (RLD) Fisher information matrix associated with $\mathrm{SU}(2)$ rotations of the qubit state. When the output qubits have higher purity, corresponding to concentration, the optimal rate is given by the ratio of the maximum eigenvalues of the input and output RLD matrices. In contrast, when the output qubits have lower purity, corresponding to dilution, the optimal rate is given by the ratio of their minimum eigenvalues. Remarkably, both concentration and dilution can be implemented using SWAP tests as the only nontrivial two-qubit measurement primitive, together with ancillary qubits initially prepared in maximally mixed states, without requiring any additional two-qubit gates. Our work thus provides a novel operational interpretation of the full complex RLD Fisher information matrix. Crucially, its antisymmetric, purely imaginary part encodes geometric information beyond the statistical distance between density operators and plays an essential role in determining the optimal state-conversion rates.

Policy learning aims to determine who should be treated based on individual characteristics. In high-stakes settings such as medicine and public policy where safety is a central concern, improving the average outcomes alone may not be sufficient: decision makers may also seek to protect individuals from harm, in line with the Hippocratic principle of “do no harm.” In this paper, we propose conformal policy learning (CPL), a policy learning procedure with a new distribution-free safety guarantee that controls the probability of assigning treatment to an individual who would be harmed relative to control. CPL views each treatment decision as testing a hypothesis of counterfactual harm and assigns treatment by thresholding conformal p-values. These p-values use observable proxies and selective calibration to address the challenge that the potential outcomes under comparison are never simultaneously observed. For randomized experiments, under standard exchangeability conditions, CPL provides finite-sample safety guarantee at a user-specified level, without imposing any outcome modeling assumptions. Moreover, when the outcome model is consistently estimated, CPL achieves asymptotically optimal welfare subject to the safety constraint. In observational studies, CPL with learn-then-balance weights achieves doubly robust safety guarantees. We evaluate CPL through extensive simulations and apply it to an empirical study of AI-powered interventions designed to reduce conspiracy beliefs.

Two features intrinsic to survey sampling complicate semiparametric efficiency analysis: design-induced dependence among sampling indicators and the randomness of finite-population targets under the superpopulation law. For general semiparametric full-data models, we show that the observed-data experiment under a broad class of dependent designs is locally asymptotically normal with the tangent-space structure of a reference Poisson experiment. Under the joint superpopulation-design law, first-order efficiency depends on the design only through the limiting inclusion-probability function. Standard missing-at-random projection in the reference experiment characterizes the observed-data efficient influence function. Finite-population targets are treated through first-order asymptotic expansions, extending the analysis beyond exact random sums to nonlinear census characteristics. Superpopulation and finite-population centerings yield equivalent notions of local regularity and efficiency, with their bounds linked by a Pythagorean decomposition that gives a generalized finite-population correction. We then give general and design-specific conditions under which cross-fitted estimators with estimated nuisance functions attain both efficiency bounds. For the finite-population mean, the bound equals the large-sample limit of the Godambe-Joshi anticipated-variance lower bound. For scalar targets, we characterize optimal limiting inclusion probabilities. Simulations and California Academic Performance Index data illustrate the theory.

In this paper we study a linear drift perturbed by a superposition of $m$ independent fractional Brownian motions with known Hurst parameters and a common scale, observed at $N$ equidistant times. Inference for such models is usually asymptotic; we show that here it is exact. We derive the maximum likelihood estimators of the drift $\theta$ and of the scale $\alpha^{2}$ in closed form and obtain their exact finite-sample joint law: $\widehat\theta$ is Gaussian, $N\widehat\alpha^{\,2}/\alpha^{2}$ is chi-square with $N-1$ degrees of freedom, and the two are independent. As this law is free of every model parameter, we deduce Student and chi-square confidence intervals and tests of exact level for every $N\ge2$, whatever the Hurst vector. We also prove that the estimators are uniformly minimum variance unbiased with $\widehat\theta$ attaining the Cramér–Rao bound at every $N$, that both are strongly consistent and asymptotically normal, and that the drift estimators form, in law, a Brownian motion run along their own variance scale. A sharp non-asymptotic bound shows that the accuracy of the drift is governed by the length of the observation window and not by the mesh, and a Monte Carlo study confirms exact coverage, even at small sample sizes, and quantifies what is lost when the Hurst vector is misspecified.

Langevin-based Markov chain Monte Carlo (MCMC) algorithms use gradient information to improve sampling, particularly in high dimensions. Classical optimal scaling theory for these algorithms has largely focused on the Metropolis-Hastings (MH) acceptance rule. However, there has been a recent surge in acceptance rules beyond MH for applications spanning differential privacy, stochastic MCMC, diffusion models, and molecular dynamics. We develop optimal scaling results for Langevin proposals employed with generalized acceptance rules belonging to a suitable class. For high-dimensional targets, we recover the usual $O(d^{-1/3})$ scaling, while different acceptance rules lead to different optimal acceptance probabilities.

For multivariate scale and location–scale models with independent components, we extend the univariate results of Zhou and Nayak (2012) and derive optimum equivariant estimators under the generalized Pitman closeness criterion. We first show, by a counterexample, that in the multivariate case the Pitman closeness comparison within the class of equivariant estimators is not transitive, so that a Pitman closest equivariant estimator does not exist in general. We then enlarge the transformation group by the coordinate permutations—equivalently, impose the formal equivariance principle of Berger (1985) across isomorphic component problems, in the spirit of the separable rules of Robbins' (1951) compound decision theory—and show that within the resulting restricted class an optimum is restored. A multivariate median lemma based on strictly convex losses then yields explicit Pitman closest equivariant estimators of the scale parameters, powers of the scale parameters, and the location parameters, given by median-adjusted versions of any given equivariant estimator. Applications to the multivariate uniform and multivariate normal distributions are worked out in detail. Monte Carlo experiments for the Rayleigh distribution and for a competing risks model with Rayleigh component lifetimes confirm that the proposed estimators dominate the maximum likelihood and Bayes estimators under the Pitman closeness criterion, and a real industrial data set on ball bearing failure times illustrates the feasibility of the method in practice.

Equivariance is increasingly used in machine learning and statistics, often without systematic justification. In a companion article, the equivariance criterion was applied to the normal linear model with a fixed design matrix (fixed-$X$), yielding the minimum risk equivariant (MRE) estimators of the coefficient vector and of the condensed diagonal covariance matrix under a multivariate invariant location–scale group. We extend these results to the random-$X$ case, with covariates sampled from a population. The extension hinges on a distinction vacuous for fixed-$X$ but fundamental for random-$X$: whether risk and unbiasedness are evaluated conditionally on the realized design or after averaging over the design distribution. Under conditional evaluation, the fixed-$X$ group applies given $X$: least squares remains the best equivariant estimator of the coefficient vector, and the MRE estimators of the population variances keep their fixed-$X$ forms with population sizes at the realized design. Under absolute evaluation with an i.i.d.\ design, the picture changes qualitatively: the natural scale group acting jointly on $(Y,X)$ fixes the coefficient vector, the induced parameter-space action is intransitive, equivariant risks are constant only along orbits indexed by the signal-to-noise ratio $\rho=\|\beta\|^2/\sigma^2$, and no uniformly minimum risk equivariant estimator exists. In the scalar case the optimal equivariant weight is the oracle shrinkage factor $w^*(\rho)=\rho/(\rho+E[T^{-1}])$, with least squares recovered as the infinite-signal limit $\rho\to\infty$—explaining and refining the known failure of the Gauss–Markov theorem with random regressors. For a centered design under location–scale transformations, least squares remains optimal within the natural invariant-contrast class, and the MRE estimator $S^2/(n-p+2)$ of the error variance is valid under both modes.

Protecting individual privacy has become a central and urgent concern in modern data analysis, given the vast quantities of data now generated and processed. In this paper, we develop a systematic theory of exact asymptotic efficiency for regular parametric estimation under central zero-concentrated differential privacy. In the privacy regime, the governing object is a diameter-constrained information region: the set of information matrices generated by statistics with diameter at most one. For weighted quadratic loss, we show that the exact local minimax risk is an inverse information variational functional over this region. More generally, a mixed information region yields a unified efficiency constant across different regimes, covering the privacy regime and classical Fisher efficiency. A matching estimator releases the empirical mean of a nearly optimal bounded statistic with Gaussian noise and locally inverts its population moment map. Our theory differs from classical efficiency theory in its set-valued information geometry and loss-dependent efficient estimator. As examples, we provide closed-form constants and optimal procedures for various concrete models, including one-dimensional regular families, Gaussian means, categorical probability vectors, and regression models among others. The theory also transfers to Gaussian differential privacy through an exact parameter rescaling.

This work addresses a longstanding gap in the statistical foundations of marginal maximum likelihood estimation for high-dimensional latent variable models. Marginal maximum likelihood estimation is widely used to fit latent variable models across the social sciences, ecology, and machine learning. Despite its broad use, rigorous asymptotic theory for nonlinear models remains limited when both the sample size and the number of observed variables diverge. The gap arises largely from the fact that integration over the latent variables creates a nonlinear objective that tightly couples the high-dimensional model parameters. To address this issue, we first show that the marginal likelihood exhibits multiple nearly flat directions even at the true parameter, in contrast to the behavior in the fixed-dimensional case. Building on this geometric characterization, we develop new techniques to establish consistency and asymptotic normality for the marginal estimator of the high-dimensional parameters. For the latent variables, we provide frequentist and Bayesian uncertainty quantification, proving asymptotic normality of the maximum a posteriori estimator and a Bernstein-von Mises-type result for the plug-in posterior. Together, these results provide rigorous foundations for marginal estimation and latent-variable inference in high-dimensional models.

Parameter estimation in finite mixture models can exhibit highly heterogeneous convergence behavior: locally isolated components may be estimated substantially faster than groups of competing components. Existing analyses based on Wasserstein distances typically characterize only the worst-case rate and therefore do not fully capture this local heterogeneity. In this paper, we introduce a Voronoi-based partial optimal transport (VPOT) framework for obtaining refined local and global convergence guarantees for the maximum likelihood estimator of the mixing measure. The key geometric idea is to localize the comparison of two mixing measures to extended Voronoi neighborhoods and use partial optimal transport to accommodate the unequal masses of their local restrictions. Within each neighborhood, the first-order POT discrepancy is raised to a power determined by the number of locally competing atoms, allowing the resulting loss to adapt to the local degree of singularity. Under suitable regularity and strong identifiability conditions, we establish uniform local and global upper bounds for a maximum likelihood estimator under the VPOT loss. These bounds reveal a configuration-dependent form of parameter estimation: less singular local configurations admit faster convergence, whereas the most singular configuration recovers the classical worst-case behavior characterized by Wasserstein-based analyses. We further establish a minimax lower bound showing that the convergence rate for estimating the mixing measure under the VPOT loss is optimal. Our results hold in arbitrary fixed dimension without requiring mixing proportions to be uniformly bounded away from zero or prior knowledge of the true number of mixture components. Overall, VPOT provides a configuration-adaptive framework for capturing heterogeneous parameter-estimation behavior in finite mixture models.

Statistical precision and scientific relevance operate on different scales. In regular problems, sampling uncertainty contracts at rate $n^{-1/2}$, whereas the magnitude below which an effect is scientifically negligible may be fixed or may vary with information. Let $\Delta_n$ denote a relevance threshold and $I_0$ Fisher information, and define $\lambda_n=\sqrt{nI_0}\Delta_n$. For separated negligible and meaningful parameter classes, we establish the minimax lower bound $\liminf R_n^*\ge 2\{1-\Phi(\varepsilon\kappa)\}$ when $\lambda_n\to\kappa$. If $\lambda_n\to0$, the experiments merge and consistent classification is impossible. If $\lambda_n\to\kappa\in(0,\infty)$, the problem converges to a Gaussian-shift decision problem. We characterize the exact minimax rule and risk in that limiting composite problem. As $\kappa\downarrow0$, the optimal cutoff converges to one statistical-error unit and the optimal improvement over trivial risk is $O(\kappa^2)$, while a simple relevance-boundary rule improves only at $O(\kappa^3)$; as $\kappa$ grows, that rule becomes asymptotically minimax. If $\lambda_n\to\infty$, consistent classification is attainable under a uniform estimation-resolution condition, verified for Gaussian and Bernoulli models. For $\Delta_n=dn^{-\gamma}$, the critical rate is $\gamma=1/2$. We also show that moving asymmetric relevance regions depend on standardized distances to their two boundaries rather than total width alone. For heterogeneous true effects, we derive a significance-saturation limit and show that point-null significance can be most selective for scientifically relevant effects at intermediate information. The framework characterizes when scientific relevance is statistically resolvable through a common information scale.

We study exact community recovery in the Degree-Corrected Poisson Stochastic Block Model (DC-SBM) through the maximum profile likelihood estimator. Our main contribution is identifying an explicit complexity constant $C(\pi,S)$, which closely captures the exact recovery threshold. This constant combines a weighted Chernoff–Hellinger separation between communities with a Kullback–Leibler correction induced by the degree-normalized block volumes. We rigorously prove that $C(\pi,S)>1$ ensures the strong consistency of the maximum profile likelihood estimator, yielding exact community recovery down to the logarithmic sparsity scale. We conjecture that $C(\pi,S)=1$ represents the sharp information-theoretic threshold, providing an upper bound on the exact-recovery transition, and we discuss analogies with the binary SBM and binary DC-SBM cases. Additionally, we establish weak consistency when the average degree diverges, offering a unified picture across sparse and dense regimes. The proofs require new control of the profile likelihood in the presence of heterogeneous degree parameters, whose node-specific weights fundamentally alter the geometry and concentration of the recovery problem.

We develop a Gaussian comparison theory for posterior inference in non-Gaussian high-dimensional models. The framework allows for model misspecification and does not require the posterior distribution itself to be approximately Gaussian. Working directly with likelihood processes on separable function spaces, we establish explicit nonasymptotic free energy bounds controlled by first two moment discrepancies and local third moments. A perturbation-and-convexity argument transfers these comparisons to sandwich bounds for posterior mean squared error and posterior variance, including one-sided conclusions at nondifferentiable phase transitions. In high-dimensional product models, these conditions reduce to local moment controls and a canonical feature-radius scaling. Under weak regularity requirements, the resulting universality theory applies to a broad class of likelihood models, ranging from exponential families to more general regular local log-likelihoods. The framework is illustrated on sparse Bernoulli hypergraph inference.

Nested cubature rules, in which each refinement retains all previously used nodes and thus reuses earlier function evaluations, are natural in multilevel and adaptive integration. For every fixed $s>d/2$, we show that the equal-weight QMC integration rate on $\mathbb S^d$ is compatible with such nested point sets. In the subcritical range $d/2<s<d$, cumulative unions of geometrically growing QMC blocks yield nested QMC design sequences whose successive cardinality ratios converge to any prescribed $\rho>1$. At and above the critical index $s=d$, where the block-averaging estimate no longer yields the optimal rate, we prove an equal-weight completion theorem based on low-frequency discrepancy cancellation. A block-sensitive estimate sharpens the iteration and yields nested QMC designs for all $s\ge d$ with $N_{j+1}\lesssim(j+1)^{2s/d-1}N_j$. Every prescribed finite point set also admits an optimal-rate completion.

We develop notions of covariance and principal component analysis (PCA) for probability measures on Riemannian manifolds of bounded geometry and a discrete counterpart for distributions on the vertex sets of weighted simple graphs. Rather than anchoring variation at the Fréchet mean, whose uniqueness and usefulness can fail in this context, we consider variation about every point. This leads to a representation of each point on the manifold by a vector field derived from the heat kernel and thus to a map of the underlying manifold into a Hilbert space of vector fields, in which covariance and PCA carry over from the Euclidean setting. A distinguishing feature of this formulation is that one typically obtains infinitely many principal components, capable of capturing highly nonlinear geometric features and patterns of variation. We prove an embedding theorem for the mapping into the space of vector fields and develop two computational reductions of the theory: RiePCA, a finite-dimensional reduction based on vector fields supported on finitely many points, and GraphPCA, a discrete formulation for measures on weighted graphs in which vector fields assign orientations and magnitudes to edges. Numerical examples and experiments illustrate the behavior of the proposed methods on both synthetic and real data.

We study high-dimensional nonparametric regression on the Boolean hypercube \(\{0,1\}^d\), from a sample of size \(n\), under a Fourier–Sobolev budget, i.e., a bound \(B\) on the total quadratic influence of the regression function. The central object is the resolvable Fourier degree \(D^*(d,n)\), defined as the largest interaction order \(D\) such that the dimension of the degree-\(D\) Fourier–Walsh space does not exceed \(n\). We prove that the minimax risk over Fourier–Sobolev ellipsoids is of order \(B/D^*(d,n)\): estimation is possible up to the statistically visible interaction order, and the budget controls the residual mass carried by higher-order interactions. We then study monotone regression under the same budget. For monotone functions supported on a known set of \(s\) coordinates, with \(s\) of order \(\log n\), we obtain a matching minimax rate of order \(B/s\). Finally, we consider an unknown-support model in which the function is monotone and depends on at most \(s\) coordinates; its minimax risk is governed by the intrinsic dimension \(s\) together with the logarithmic cost of selecting the active support.

We study regression with subsets as covariates. The response is an unknown function of the input subset, and observations consist of noisy evaluations at uniformly sampled subsets, each containing exactly \(k\) items from a ground set of size \(d\). This problem arises in combination screening, bundle preference modeling, and other settings in which outcomes depend on collections of prescribed size. The fixed-cardinality constraint couples the membership coordinates, so standard product-domain notions of interaction and smoothness cannot be imported unchanged. We use the harmonic decomposition of the Johnson graph to define an intrinsic Johnson–Sobolev scale and determine the exact dimensions of low-order interaction spaces. We derive finite-sample risk bounds for harmonic projection and establish a nonasymptotic minimax characterization when the Johnson–Sobolev energy budget is controlled relative to the noise variance. We also show that stable least squares removes the signal-dependent fluctuation of empirical projection when the empirical Gram matrix is sufficiently well conditioned. For arbitrary signal-to-noise ratios, a rank-deficiency analysis quantifies the components left unidentified by the random design. Together, the rank-deficiency analysis and a centered completion estimator yield minimax bounds that match up to constants whenever \(\min(k,d-k)\) and the smoothness order are fixed. Numerical experiments reported in the Supplementary Material illustrate estimation under different interaction profiles and distinguish the effects of observation noise, random-design fluctuation, and incomplete coverage.

We extend the Ibragimov-Khasminskii likelihood ratio analysis method, in the one-dimensional parameter case, to a framework based on the Skorokhod $M\_1$ topology. The proposed framework applies to a broad class of statistical models, including those for which the normalized likelihood ratio processes are continuous while the limiting likelihood ratio process is discontinuous, a setting outside the scope of classical approaches based on the uniform or Skorokhod $J\_1$ topologies. We derive sufficient conditions for the weak convergence of likelihood ratio processes in the space of càdlàg functions on $\RR$ vanishing at $\pm\infty$, endowed with the Skorokhod $M\_1$ topology, and show how this convergence yields the asymptotic behavior of the maximum likelihood estimator. We also introduce new techniques for controlling the $M\_1$ modulus of continuity. Although these techniques are particularly well suited to models in which all jumps and steep continuous transitions are in the same direction, they are of independent interest and may prove useful beyond this setting. As an application, we study a smooth change-point model for inhomogeneous Poisson processes in the fast regime, where the transition interval shrinks faster than $1/n$. We establish that in this case the maximum likelihood estimator has the same asymptotic behavior (consistency, rate of convergence, limiting distribution and convergence of moments) as in the corresponding ”pure” change-point model.

Fraction-based information measures have received considerable attention for describing complex systems, as they enable the investigation of signals with high sensitivity \citep{machado2014fractional}. In this paper, we introduce quantile versions of fractional cumulative past inaccuracy (FCPI) and dynamic fractional cumulative past inaccuracy (DFCPI) measures based on the inverse Mittag-Leffler function (MLF) or fractional logarithm function, which are the extensions of the fractional cumulative past and dynamic fractional cumulative past entropies, respectively. The various properties of quantile-based FCPI and its dynamic version are provided. We also propose a nonparametric estimator for the proposed measure, and simulation studies are carried out for validation. Finally, we bring out real data application of the newly introduced quantile-based FCPI.

We consider nonparametric regression with a Banach-space-valued predictor where the covariate takes values in a general separable Banach space. Our primary objective is to develop a unified framework for estimating the regression operator without relying on an inner-product structure, thereby extending classical and functional nonparametric regression beyond Euclidean and Hilbert spaces in the presence of measurement error. We establish the fundamental properties of the proposed estimator and develop its large-sample theory under conditions formulated in terms of the geometry and local concentration of the Banach-valued predictor. In particular, we study pointwise estimation and inference and discuss the construction of pointwise confidence intervals and uniform confidence bands. Our proposed framework is further extended to accommodate error-in-variables settings, where the predictor is observed with contamination. We also investigate related inverse problems arising from the construction of inverse weighting functions and consider partial contamination models. These developments provide a general framework for nonparametric regression and inference with complex infinite-dimensional predictors and identify several statistical and mathematical challenges that arise when the usual Hilbert-space structure is unavailable.

We study sampling from a distribution supported on an unknown compact $d$-dimensional $C^2$ manifold $M\subset\mathbb{R}^D$, observed only through i.i.d. uniform points from $M$. We reconstruct the constraint using an adaptive local-convex-hull estimator and target an ambient distribution penalized by squared distance to the reconstruction. Although the reconstructed set may be nonsmooth or fail to be a manifold, its Hausdorff accuracy alone suffices to control the Wasserstein error. We quantify the tradeoff between reconstruction accuracy and penalty strength and show that optimal tuning gives an error proportional to the square root of the Hausdorff error. A planar example proves that this dependence is sharp for the proposed scheme. For a $d$ dimensional constraint reconstructed from $N$ observations, the error becomes $\mathcal{O}( (N/N)^{1/d})$. Finally, we prove uniform geometric ergodicity of a Gaussian random-walk Metropolis–Hastings sampler and combine reconstruction, approximation, and mixing into an explicit finite-time guarantee.

Let $P_d$ be the elliptope of $d \times d$ correlation matrices, and, for a standardised law $F$, let $S_d^F$ be the set of correlation matrices of random vectors with all margins $F$. We give two short proofs. First, an eleven-term representation of $|x|^4$ as a sum of fourth powers of linear forms on $\mathbb{R}^4$ yields $S_{11} \neq P_{11}$ for uniform margins. Second, the six diagonals of the regular icosahedron yield $S_6^F \neq P_6$ for arcsine margins. Together with a result of Devroye and Letac, this gives $S_d^F = P_d$ if and only if $d \le 5$, settling a conjecture of theirs.

Heavy-tailed response variables are routinely modeled as a function of covariates, but the asymptotic theory available for the resulting maximum likelihood estimators sidesteps certain technical difficulties by restricting the parameters to a compact set, which is not the natural parameter space. We consider a Fréchet model whose scale parameter depends log-linearly on a vector of covariates, while the tail parameter is constant, and we study the conditional maximum likelihood estimator of the tail parameter and the regression coefficients jointly. We show that this estimator is consistent over the full, noncompact parameter space, and that it is asymptotically normal, with the inverse Fisher information as limiting covariance matrix. Consistency is obtained by a specific compactification of the parameter space relying on the cosmic closure of Euclidean space. The extension of Wald's consistency theorem needed for that step, from single observations to blocks of observations, is stated in an abstract setting and is possibly of independent interest.

We introduce an equivalence relation on exponential families, and prove that two exponential families are equivalent in this sense if and only if the corresponding induced Hessian manifolds are isomorphic. Moreover, we classify exponential families of order 1 with constant Hessian sectional curvature. To this end, we show that for an exponential family of order 1, it has constant Hessian sectional curvature if and only if the natural exponential family equivalent to the given family has a quadratic variance function (NEF-QVF). The classification coincides with that of NEF-QVF by Morris (1982) essentially.

A covariance matrix rarely distorts in a single direction: a shift can expand all variances simultaneously, shift variance along one or two principal directions, displace spectral mass without changing marginal means, or rotate the dependence structure. Default mean-shift detectors are blind to these effects, and no existing online procedure calibrates a family of spectral deviation tests with provable false alarm control across both time and tracking window selection; this paper fills that gap. We consider four spectral deviations: $D_{KL}(P_{1}\|P_{0})$, $D_{KL}(P_{0}\|P_{1})$, Jeffreys, and Bhattacharyya, each evaluated on the eigenvalues of the empirical relative covariance operator $\widehat{\Sigma}_{0}^{-1/2}\widehat{\Sigma}_{t}\widehat{\Sigma}_{0}^{-1/2}$. Critical values come from a conditional parametric bootstrap that accounts for estimation uncertainty in both the past and tracking windows, an aspect asymptotic approaches typically overlook. The procedure controls the false alarm rate family-by-family over a predefined monitoring period and a range of candidate window sizes; when a single operational window is needed, a power-based criterion selects it. We prove consistency under constant alternatives via local spectral expansions with second-order sensitivity near the null. Simulations are conservative under the null and show detection power depends largely on spectral shape rather than magnitude: $D_{KL}(P_{1}\|P_{0})$ excels under global inflation, while Jeffreys and Bhattacharyya cover a broader range of alternatives. We illustrate the approach on three financial applications: European stock indices, Fama-French sector portfolios, and large-cap technology stocks.

Retrieval-Guided Fine-Tuning (RAG-FT) incorporates retrieved data directly into the training objective, but the statistical consequences of noisy retrieval during training remain theoretically undercharacterized. We study this question by modeling RAG-FT as an estimation problem in a multi-task linear regression framework, using an OLS proxy for single-layer linear self-attention to obtain finite-sample risk bounds. Under homoscedastic retrieval noise, we show that retrieval failure decays exponentially with task separation relative to noise, and derive explicit finite-sample conditions under which RAG-FT achieves lower risk than both target-only and full-corpus training. We then introduce a Distance-Proportional Noise (DPN) model, in which retrieval quality degrades with rank, and compare two estimators under the same retrieval process: the OLS proxy and the literal, uniform-weight forward pass of linear self-attention. We prove that the attention estimator's bias diverges as $\Theta(n^{2q})$ even under exact retrieval, while OLS risk remains $\Theta(d/n)$ for every noise exponent $q>0$. These results locate the instability not in noisy retrieval itself, but in the fixed, unweighted aggregation of the literal LSA forward pass, which reweighting by reliability empirically removes. We validate the predicted rate separation through direct simulation of the DPN model.

We study the asymptotic null spectrum of Gaussian-weighted tests of univariate normality with estimated location and scale. Closed Hermite coefficients give equations both away from and at the unperturbed poles. Every positive eigenvalue of the fully standardized covariance is simple, and the odd and even eigenvalues alternate strictly. For each unperturbed even pole except the largest, there is exactly one weight parameter at which it belongs to the perturbed even spectrum. These parameters are strictly ordered and converge to the boundary of the parameter interval. A signed total-positivity argument proves simplicity more generally for consecutive Gaussian covariance corrections and positive even integrable weights. We also derive the limiting quadratic forms directly from degenerate kernels at standardized observations. A derivative summability condition controls the complete spectral tail, including signed kernels, and is verified for characteristic-function kernels with a finite fourth weight moment. A real Hilbert-space factorization identifies the observation and Fourier spectra. Numerical calculations assess asymptotic calibration.