For mean-zero unit-variance random variables with fourth moment at most κ, the sharp one-sided tail V₁(t,κ) is completely mapped into four explicit regimes, with matching certificates and a proof-degree phase transition.
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Billingsley,Convergence of Probability Measures
23 Pith papers cite this work, alongside 14,151 external citations. Polarity classification is still indexing.
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Necessary and sufficient conditions for convergence of low-intensity Poisson–Voronoi diagrams to a unique ideal tessellation, applied to symmetric spaces and Diestel–Leader graphs.
Function graph transformers use graph measures to provide a measure-theoretic framework where standard transformer components universally approximate operators between function spaces while preserving single-valued function outputs.
A multi-time Markov renewal theory on partially ordered lattices is developed, yielding stratified inverse-renewal limits that are Gaussian on single-coordinate cells and non-Gaussian minima on interfaces, plus exact-time local theorems and Markovian augmentation criteria.
Introduces downward conditional monotonicity for MMPP to obtain stochastic domination bounds that determine survival and extinction regimes for contact processes in finite-state random environments via QBD eigenvalue comparison.
The fast-reversion limit of Heston is an interval-valued process whose excursions are invisible to vanilla options but raise touch-option prices by up to ~10%.
Derives matching lower and upper bounds on minimal detectable intra-block drift amplitude δ_min(n,α,β)=Θ(n^{-1/2}) for finite-key E91-type QKD using minimax hypothesis testing and CUSUM statistic.
Profile MLE for the regime-switching threshold in null-recurrent diffusion converges at rate n^{-(1+γ)/2} to the arg sup of a doubly stochastic drifted Poisson process involving local time of oscillating Brownian motion.
For polynomially mixing billiards with cusps, Birkhoff sums of observables φ(x) = d(x,x0)^{-2/α} with tail index α satisfy stable laws whose index is a function of both α and the mixing exponent γ when γ ∈ (1/2,1) and α ∈ (0,2) excluding 1.
Generalized entanglement entropies are constructed via left-, right-, and bi-invariant unit-invariant singular value decompositions to ensure scale invariance for non-Hermitian and rectangular operators in quantum mechanics, random matrices, and Chern-Simons theory.
For fixed retention probability θ, the process log L_⌊nt⌋ obeys a functional LDP with geometric-mark entropy rate, an MDP with the CLT Gaussian RKHS rate, and a Strassen LIL with that unit ball as cluster set.
Defines resilience evaluation D^ρ π as the L1-limit of scaled dynamic risk measure applied to process increments, and derives its dual representation as worst-case conditional expectation of an effective drift when ρ arises from BSDEs with Lipschitz or quadratic drivers.
Defines the H_α family of balance indices for phylogenetic networks, establishes structural properties including a grafting property, and analyzes minima, maxima, and distributions under random models such as Yule and PDA.
Characterizes duals of white-noise-driven continuous stochastic flows by explicit SDEs and introduces a self-dual polynomially self-repelling flow model.
The IM interval is the shortest valid prior-free procedure for the Behrens-Fisher problem, established via cylindrical predictive random sets, minimaxity, admissibility, and a projection argument.
Develops a unified framework representing performance metrics as smooth functionals of confusion-matrix probabilities to enable cluster-robust sandwich variance estimation for asymptotically valid confidence intervals and tests under clustered data.
DMW is a scalable Wasserstein statistic over random distance-matrix laws that provably lower-bounds and converges to Gromov–Wasserstein.
Proves well-posedness and unique invariant measure (delta at 0 plus gamma density) for sticky CIR, derives Green's function for exact sampling, and analyzes MH and ULA samplers with explicit bias for the potential case via Girsanov.
Establishes weak convergence of the quadratic field for speed-change Kawasaki dynamics to equilibrium fluctuation in the non-gradient case.
Strengthens classical scaling limit theorems for correlated random walks to functional convergence in Hölder and rough Hölder topologies.
Proves E[max_j | (1/n) sum_i ε_ij |] ≥ min{255/256, (1/sqrt(2 log 2)) sqrt(log(2p)/n)} with equality for (n,p)=(2,1) and (2,8).
The body of this submission derives (with gaps) a β-mixing U-statistic limit for a spatial Cramér–von Mises test, while the abstract promises a different general-dimensional origin-invariant statistic that never appears in the text.
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The Exact Worst-Case Tail Probability under Bounded Kurtosis
For mean-zero unit-variance random variables with fourth moment at most κ, the sharp one-sided tail V₁(t,κ) is completely mapped into four explicit regimes, with matching certificates and a proof-degree phase transition.
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Convergence towards Ideal Poisson--Voronoi tessellations with a focus on Diestel--Leader graphs
Necessary and sufficient conditions for convergence of low-intensity Poisson–Voronoi diagrams to a unique ideal tessellation, applied to symmetric spaces and Diestel–Leader graphs.
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Function graph transformers universally approximate operators between function spaces
Function graph transformers use graph measures to provide a measure-theoretic framework where standard transformer components universally approximate operators between function spaces while preserving single-valued function outputs.
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Multi-time Markov renewal chains and stratified renewal theorems
A multi-time Markov renewal theory on partially ordered lattices is developed, yielding stratified inverse-renewal limits that are Gaussian on single-coordinate cells and non-Gaussian minima on interfaces, plus exact-time local theorems and Markovian augmentation criteria.
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Downward conditional monotonicity gives survival and extinction for contact processes in random environments
Introduces downward conditional monotonicity for MMPP to obtain stochastic domination bounds that determine survival and extinction regimes for contact processes in finite-state random environments via QBD eigenvalue comparison.
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Fast-excursion limit of the Heston model
The fast-reversion limit of Heston is an interval-valued process whose excursions are invisible to vanilla options but raise touch-option prices by up to ~10%.
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Detectability Limits for Intra-Block Temporal Drift in Finite-Key Entanglement-Based QKD
Derives matching lower and upper bounds on minimal detectable intra-block drift amplitude δ_min(n,α,β)=Θ(n^{-1/2}) for finite-key E91-type QKD using minimax hypothesis testing and CUSUM statistic.
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Self-organized regime switching in null-recurrent dynamics
Profile MLE for the regime-switching threshold in null-recurrent diffusion converges at rate n^{-(1+γ)/2} to the arg sup of a doubly stochastic drifted Poisson process involving local time of oscillating Brownian motion.
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Stable laws for heavy-tailed observables on polynomially mixing billiards
For polynomially mixing billiards with cusps, Birkhoff sums of observables φ(x) = d(x,x0)^{-2/α} with tail index α satisfy stable laws whose index is a function of both α and the mixing exponent γ when γ ∈ (1/2,1) and α ∈ (0,2) excluding 1.
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Generalised Entanglement Entropies from Unit-Invariant Singular Value Decomposition
Generalized entanglement entropies are constructed via left-, right-, and bi-invariant unit-invariant singular value decompositions to ensure scale invariance for non-Hermitian and rectangular operators in quantum mechanics, random matrices, and Chern-Simons theory.
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Functional Limit Theorems for Random Least Common Multiples
For fixed retention probability θ, the process log L_⌊nt⌋ obeys a functional LDP with geometric-mark entropy rate, an MDP with the CLT Gaussian RKHS rate, and a Strassen LIL with that unit ball as cluster set.
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Financial Resilience Evaluation: From Conditional Expectations to Dynamic Convex Risk Measures
Defines resilience evaluation D^ρ π as the L1-limit of scaled dynamic risk measure applied to process increments, and derives its dual representation as worst-case conditional expectation of an effective drift when ρ arises from BSDEs with Lipschitz or quadratic drivers.
-
A parameterized family of balance indices for phylogenetic networks
Defines the H_α family of balance indices for phylogenetic networks, establishes structural properties including a grafting property, and analyzes minima, maxima, and distributions under random models such as Yule and PDA.
-
Continuous stochastic flows driven by white noise and their duals
Characterizes duals of white-noise-driven continuous stochastic flows by explicit SDEs and introduces a self-dual polynomially self-repelling flow model.
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Revisiting the Behrens-Fisher Problem: Validity-First Optimality
The IM interval is the shortest valid prior-free procedure for the Behrens-Fisher problem, established via cylindrical predictive random sets, minimaxity, admissibility, and a projection argument.
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Beyond Point Estimates: Reliable Evaluation of Prediction Performance Metrics under Clustered Data
Develops a unified framework representing performance metrics as smooth functionals of confusion-matrix probabilities to enable cluster-robust sandwich variance estimation for asymptotically valid confidence intervals and tests under clustered data.
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Distance-Matrix Wasserstein Statistics for Scalable Gromov--Wasserstein Learning
DMW is a scalable Wasserstein statistic over random distance-matrix laws that provably lower-bounds and converges to Gromov–Wasserstein.
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Sticky CIR process with potential: invariant measure and exact sampling
Proves well-posedness and unique invariant measure (delta at 0 plus gamma density) for sticky CIR, derives Green's function for exact sampling, and analyzes MH and ULA samplers with explicit bias for the potential case via Girsanov.
-
Quadratic fluctuations of speed-change Kawasaki dynamics
Establishes weak convergence of the quadratic field for speed-change Kawasaki dynamics to equilibrium fluctuation in the non-gradient case.
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Functional Scaling Limits of Interpolated Correlated Random Walks in H\"older Topology
Strengthens classical scaling limit theorems for correlated random walks to functional convergence in Hölder and rough Hölder topologies.
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Notes on constants for maxima of Rademacher averages
Proves E[max_j | (1/n) sum_i ε_ij |] ≥ min{255/256, (1/sqrt(2 log 2)) sqrt(log(2p)/n)} with equality for (n,p)=(2,1) and (2,8).
-
Weak Convergence and Gaussian Limits for a General-Dimensional Origin-Invariant Cram\'er--von Mises Statistic
The body of this submission derives (with gaps) a β-mixing U-statistic limit for a spatial Cramér–von Mises test, while the abstract promises a different general-dimensional origin-invariant statistic that never appears in the text.
- Invariant Measures and Weak-Magic-Injection Asymptotics in Random Monitored Quantum Circuits