Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics · Classical and Real Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
For independent copies of a symmetric variance-one input, we study entropy of their variance-preserving weighted sum after Gaussian smoothing. For every Q < 1/2, we prove all-weight strict entropy concavity for 0 < q <= Q in an explicit infinite-dimensional Gaussian-relative bounded-density neighborhood. A uniform moment envelope with finite nonzero Hermite discrepancy gives a common eventual threshold; every fixed symmetric non-Gaussian all-moment input has eventual all-weight strict concavity. For the symmetric log-concave class, any counterexample sequence as q decreases to zero must converge weakly and in each fixed absolute moment to the Gaussian, while endpoint curvature is strictly negative for q <= 1/20. An exact moment-matched exponential-tail construction delineates the remaining near-Gaussian gap. A common all-weight threshold for the full symmetric log-concave class remains open.
Edition 6
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
This paper compares a homogeneous real-stable count law with its mean-matched multinomial law through the normalized covariance deficit D. Finite chi-square bounds retain Tr(D) and Tr(D²), with an exact covariance-determined quadratic likelihood component and a uniform remainder. Matching mean and covariance gives a sharp total-variation rate of order Tr(D)^{3/2}. The optimal centered real elementary-symmetric coefficient inequality is proved in every order with its equality classification. A sharp centered complex degree-five coefficient bound in six variables yields the optimal factor 3/4 for the fifth likelihood component of every degree-six stable count law, at arbitrary covariance rank. Together with the other orders, this establishes the complete-homogeneous factor-one target throughout degree six. General degrees retain a factor-24 envelope, with stronger order-five bounds and structured spectral regions. Higher-interior factor one in degrees at least seven remains open. A fixed-law Gaussian mutual-total-variation limit is retained. Classical root-extremum, determinantal, centered-coefficient and orthogonal-expansion methods are credited; standalone code reconstructs finite exact certificates.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics
- Paper type
- Methods Paper
- Manuscript stage
- Preprint
We study the normalized covariance deficit of a homogeneous real-stable count law on m categories with total count k. This deficit is a rescaling of an established canonical dependence matrix for two balls drawn without replacement. Its positive semidefiniteness gives a diagonal domination property that controls rare categories and normalization error. We use this structure to construct an upper confidence bound for its operator norm without a minimum category-probability assumption and without knowing the mean. Independent batches of full count observations suffice: the mean stage uses O(m(k − 1)ε^{−2} log(m/δ)/k) observations and the second stage uses O(m(k − 1)²ε^{−2} log(m/δ)). A stable degree-two subclass gives a matching linear category and inverse-square accuracy lower bound up to logarithms. Real-rooted one-dimensional projections then turn a covariance bound and a reference Fourier margin into simultaneous positive-parity certificates. We give a sharp positivity boundary on a binary subclass and stable laws with identical means and covariance but opposite parity. The constants in the confidence bound are conservative, and the observation model is classical labeled counts.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
We estimate fixed smooth pullback entries of the inverse weighted-Laplacian metric of an unknown smooth strictly positive density on a flat torus. The estimator uses symmetric heat-kernel Sinkhorn normalization of the sample and prescribed score functions, without density evaluations. For every fixed dimension we establish a joint, population-centered root-n central limit theorem as the heat bandwidth tends to zero. The proof gives a signed-measure Taylor expansion on one Sobolev space, an explicit influence function including normalization and inverse response, and a consistent empirical covariance estimator using exact solves. A scalar Hessian identity with the moving integration measure improves the sufficient remainder budget. The population tensor has even finite asymptotic jets; fixed multiband extrapolation removes enough bias to retain the same limiting covariance. The influence is the canonical gradient, and the estimator attains the pointwise information bound under smooth local density submodels. Finally, an observed scaling, spectral-gap and inverse-residual certificate produces an actual finite first-hit output with negligible root-n metric error. Its certified-arithmetic realization requires effective convergent input enclosures. Bandwidth conditions are conservative; no practical runtime, optimal smoothing rate or numerically certified covariance algorithm is asserted.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
An exact sampler reads independent fair bits and emits an iid discrete target law, retaining only a fixed finite state between outputs. Temporary computation and the depth of each round’s prefix tree are unrestricted. We prove that zero asymptotic expected information loss is possible, on finite or countably infinite support, exactly when every positive mass is a reciprocal integer power of two. The formulation uses nonnegative finite-path loss and therefore also covers infinite entropy. The countable necessity argument reduces infinitely many output kernels to finitely many integer multiplicity types, then produces an exact positive rational normalization.
For finite targets we develop a signed-charge cycle bound that remains effective when zero-gap products defeat every marked-symbol bound. For every fixed finite law with algebraic masses outside the reciprocal-power class, it gives a polynomial lower bound on loss in the number of retained states. Together with an existing exact-sampling construction, this determines logarithmic order of the minimum persistent memory as the tolerated loss tends to zero. These bounds transfer uniformly to countable conditional dyadic refinements, including infinite-entropy examples. An established logarithmic-form estimate gives a concrete exponent for a fully resonant example. Optimal leading constants and state-count exponents remain open.
Edition 2
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study the actual first residual crossing of the symmetric half-Sinkhorn iteration for observations from a fixed positive C⁸ density on the three-dimensional torus. At two heat scales, computed from the same sample, a linear combination removes the first-order population score bias. For ε = n^(−κ) with 2/17 < κ < 4/9 and every fixed 0 < γ < 3/4, stopping each run at its first projective mass residual below n^(−1/2) h^(−γ) preserves the corrected estimator’s fixed-query Gaussian limit. The stopping index is asymptotic to (3/4 − γ) log(1/h)/log 2, and the attained residual lies between c times the threshold and the threshold with probability tending to one for every fixed 0 < c < 1/2. The covariance retains the dependence between the two scales. An explicit cubic population coefficient gives a fixed analytic density for which the limit is shifted at κ = 2/17 and the normalized bias diverges below it, also under actual stopping. The formal Gaussian range extends to κ < 4/9; consistency and shrinking uncertainty require κ < 2/5. A sharp heat-spectral trace determines logarithmic corrections to the stopping count and phase-dependent residual limits, proving that the constants 1/2 and 1 in the residual bracket are sharp. Necessity or equality at the upper exponent is not asserted.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study observable residual stopping for the half-self-Sinkhorn iteration with samples from a fixed positive smooth density on the one- or two-dimensional torus. Under stated bandwidth and tolerance conditions, the first residual hit preserves the pointwise Gaussian limit of the exactly scaled empirical score. We determine the leading stopping count, bound the attained residual, and give an obstruction at the upper tolerance exponent. The proof controls the common empirical trajectory and nonuniform initialization.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study how numerical errors in symmetric kernel scaling affect the spatial score obtained from its smooth kernel extension. A full square-root-mass residual gives a sharp kernel-energy bound for every positive candidate scaling, including singular positive semidefinite Gram matrices. For Gaussian kernels this yields computable, scale-invariant pointwise score certificates. For iid samples from a strictly positive C³ density on the d-dimensional torus, we prove a simultaneous nonlinear score bound and determine the exact local modulus in a shrinking-bandwidth regime, with leading constant C_d/√f(x). Under the stated normalization and sampling assumptions, residual o(1/√n) suffices for negligible normalized score error; positive sample-dependent candidates with residual asymptotic to c/√n attain nonzero normalized error. The sufficient sampling regime is not claimed optimal. A reproducible solver experiment illustrates the certificate on actual iterates. These numerical-accuracy results do not assert a new statistical central limit theorem or an iteration lower bound.