Paper proves CLTs for Q-learning with asynchronous updates.
problem Establishing convergence rates for Q-learning algorithms.
method Polyak-Ruppert averaging, non-asymptotic and functional CLTs.
result Convergence rates in Wasserstein distance for Q-learning.
Proves CLT for Brownian paths on pinched negative curvature manifolds.
problem Distribution of Brownian paths on pinched negative curvature manifolds.
method Proof of central limit theorem for distances and Green functions.
result Central limit theorem holds for Brownian paths in pinched negative curvature.
Central limit theorem for Green metrics on hyperbolic groups.
problem Proving a central limit theorem for Green metrics on hyperbolic groups.
method Proving a central limit theorem for Green metrics on hyperbolic groups using probability measures and ordering elements.
result Proved a central limit theorem for Green metrics on hyperbolic groups.
Stochastic gradient descent's long-term fluctuations are described by a diffusion limit.
problem Long-term behavior of stochastic gradient descent in non-smooth settings.
method Functional central limit theorem applied to rescaled trajectory of SGD.
result Characterization of long-term fluctuations around the minimizer.
The study of random walks on hyperbolic spaces and Teichmüller spaces, proving central limit theorems and geodesic tracking.
problem Analyzing random walks on hyperbolic and Teichmüller spaces.
method Proving central limit theorems and geodesic tracking using finite moments and logarithmic moments.
result Translation lengths of random isometries satisfy a central limit theorem if and only if the random walk has finite second moment.
Classifies central extensions for area-preserving diffeomorphisms and shows they are fuzzy sphere limits.
problem Classifying central extensions for area-preserving diffeomorphisms.
method Classifying central extensions and showing they are fuzzy sphere limits of Kac-Moody cocycles.
result Central extensions are fuzzy sphere limits of Kac-Moody cocycles for large k.
Study shows central limit theorem for counting measures in non-smooth spaces.
problem Counting measures in non-smooth spaces with coarse negative curvature.
method Established central limit theorems for actions of groups on hyperbolic spaces without properness or smoothness assumptions.
result General framework allows for applications in geometrically finite manifolds and intersection numbers.
The study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.
problem Understanding statistical properties of zeros of random holomorphic sections.
method Proves a central limit theorem for smooth linear statistics of zero divisors of Gaussian sections in line bundles over Kähler manifolds.
result Derives first-order asymptotics and upper decay estimates for Bergman kernels.
Functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting
problem Fluctuations of boosting processes around their deterministic limit
method Stochastic perturbation analysis of ODEs in Banach spaces
result Rescaled deviations converge to a Gaussian process
We obtain a Central Limit Theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin's Omnibus Central Limit Theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the ma…
We rigorously prove a central limit theorem for neural network models with a single hidden layer. The central limit theorem is proven in the asymptotic regime of simultaneously (A) large numbers of hidden units and (B) large numbers of stochastic gradient descent training iterations. Our result describes the neural net…
This paper strengthens the central limit theorem for order statistics using relative entropy.
problem Establishing a stronger mode of convergence for central limit behavior of order statistics.
method Using relative entropy to ensure a stronger mode of convergence for central limit behavior of order statistics.
result An order O ( 1 / n ) O(1/\sqrt{n}) O ( 1/ n ) rate of convergence is established under mild conditions. Study shows mass distribution of random holomorphic sections follows a central limit theorem.
problem Understanding mass distribution of random holomorphic sections.
method Proved a central limit theorem for mass distribution of random holomorphic sections associated with positive line bundles.
result Almost every sequence of random holomorphic sections exhibits quantum ergodicity.
We prove a central limit theorem for the length of closed geodesics in any compact orientable hyperbolic surface. In the special case of a hyperbolic pair of pants, this settles a conjecture of Chas-Li-Maskit.
Study shows superdiffusive behavior in geodesic flows on curved surfaces.
problem Understanding the statistical behavior of geodesic flows on curved surfaces.
method Proved nonstandard central limit theorem with superdiffusive normalisation ( t log t ) 1 / 2 (t\log t)^{1/2} ( t log t ) 1/2 for geodesic flows on nonpositively curved surfaces. result Geodesic flows exhibit superdiffusive behavior with correlations decaying at rate t − 1 t^{-1} t − 1 . Study shows a central limit theorem for random coverings of manifolds with nilpotent groups.
problem Understanding the distribution of connected components in random coverings of manifolds with nilpotent fundamental groups.
method Used sampling homomorphisms from the fundamental group into the symmetric group and subgroup growth zeta functions of nilpotent groups.
result Proved a central limit theorem for the number of connected components of these random coverings.
We prove a central limit theorem for the components of the largest eigenvectors of the adjacency matrix of a finite-dimensional random dot product graph whose true latent positions are unknown. In particular, we follow the methodology outlined in \citet{sussman2012universally} to construct consistent estimates for the …
Given a (smooth) complex analytic family of compact complex manifolds, we prove that the central fibre must be Moishezon if the other fibres are Moishezon. Using a "strongly Gauduchon metric" on the central fibre whose existence was proved in our previous work on limits of projective manifolds, we show that the irreduc…
The study proves a quantitative functional CLT for neural networks with smooth activation functions.
problem Understanding the convergence rates of neural networks with different activation functions.
method Functional versions of the Stein-Malliavin approach and a quantitative functional central limit theorem.
result Rates of convergence depend on the smoothness of the activation function, ranging from logarithmic to sqrt(n).
We compute the central invariants of the bihamiltonian structures of the constrained KP hierarchies, and show that these integrable hierarchies are topological deformations of their hydrodynamic limits.
A new subdivision scheme for Heisenberg group values with central smoothness loss.
problem Regularity of limit curves in Heisenberg group-valued subdivision schemes.
method Interpolatory subdivision scheme with central correction based on group law.
result Central part of limit curve converges to a continuous limit with logarithmic modulus of continuity.
Study on volumes of random inscribed polytopes in projective geometries.
problem Estimating volumes of random inscribed polytopes in projective geometries.
method Central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
result Established central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
We give conditions under which the normalized marginal distribution of a semimartingale converges to a Gaussian limit law as time tends to zero. In particular, our result is applicable to solutions of stochastic differential equations with locally bounded and continuous coefficients. The limit theorems are subsequently…
The paper studies empirical processes from nearest neighbors in regression.
problem Estimating conditional cumulative distribution functions and local linear regression.
method Uniform central limit theorem and non-asymptotic bound under local bracketing entropy and uniform entropy numbers.
result Gaussian limit of empirical process with simple covariance.
NAST generalizes scattering transform for non-stationary time series analysis.
problem Analyzing non-stationary time series data.
method Neural activation of scattering transform with various activation functions and high pass filters.
result Central and non-central limit theorems for NAST of Gaussian processes.
Study smooth linear statistics on random covers of hyperbolic surfaces, showing central limit and variance results.
problem Analyzing fluctuations and energy variance of random covers of compact hyperbolic surfaces.
method Examining fluctuations in a small energy window around a fixed energy level, considering the variance of a typical surface, using a double limit where n n n and L L L go to infinity. result Distribution of fluctuations tends to a Gaussian with variance of GOE/GUE, and energy variance of a typical random n n n -cover is that of GOE/GUE. The paper examines how loss aversion impacts multi-armed bandit decisions over long periods.
problem The impact of loss aversion on multi-armed bandit decisions over long periods.
method A new central limit theorem for measures with history-dependent variances, derived under risk aversion in gains and risk loving in losses.
result Consequences of loss aversion for asymptotic properties are derived in analytical results.
Study small-time CLTs for stochastic Volterra equations with various kernels.
problem Understanding the behavior of stochastic Volterra equations with different kernels.
method Proved convergence of finite-dimensional distributions, functional CLT, and limit theorems for smooth transformations.
result Derived asymptotic pricing formulae for digital calls in rough volatility models.
For α ∈ ( 1 , 2 ) α\in (1,2) α ∈ ( 1 , 2 ) , we present a generalized central limit theorem for α α α -stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate for partial integro-differential equations (PIDEs). A classical generalized central limit theorem is recovered as a special case, p…
Study shows normal distribution in divisor counts of random sections on complex manifolds.
problem Distribution of divisors on complex manifolds.
method Central limit theorem for smooth linear statistics of Gaussian sections.
result Asymptotic normality of divisor counts.
This paper proves a central limit theorem for differential privacy in high dimensions.
problem Understanding optimal noise distributions for privacy-accuracy trade-offs in high-dimensional settings.
method Developed a central limit theorem approach to analyze differential privacy mechanisms.
result Gaussian mechanisms achieve the optimal privacy-accuracy trade-off in high dimensions.
Central bank optimizes bailout cash injection to limit defaults.
problem Optimizing cash injection to limit defaults in a system of mutual obligations.
method Proved convergence and solved a drift controlled Stefan problem using mean field control and policy gradient methods.
result Optimal strategies involve subsidizing banks with equity values in a time-dependent region.
Paper derives CLT for Bayesian neural networks trained with variational inference.
problem Analyzing the fluctuation behavior of Bayesian neural networks trained with different variational inference schemes.
method Rigorous derivation of CLT for three variational inference schemes: idealized, Bayes-by-Backprop, and Minimal VI.
result Minimal VI scheme has larger variances but is more computationally efficient.
Conservative SPDEs emerge from fluctuating SGD dynamics in neural networks.
problem Understanding the convergence of stochastic gradient descent to SPDEs.
method Mean-field analysis and central limit theorem for SPDEs.
result Optimal convergence rates for SPDEs derived from SGD.
The paper establishes a central limit theorem for estimating the influence parameter in a partially observed Hawkes process system.
problem Estimating the influence parameter in a partially observed Hawkes process system.
method Central limit theorem applied to an estimator of the influence parameter in a partially observed system of Hawkes processes.
result Establishes a central limit theorem for the estimator of the influence parameter under the subcritical condition.
Paper proves a Central Limit Theorem for Random Forest Permutation Importance Measure.
problem Lack of theoretical analysis of Random Forest Permutation Importance Measure (RFPIM).
method Formal proof using U-Statistics theory, deviating from conventional Random Forest model.
result Established a Central Limit Theorem for RFPIM.
Develops CLTs for Markov chain transition probabilities and policies.
problem Estimating transition probabilities and policies in controlled Markov chains.
method Non-parametric estimator for transition matrices; CLTs for value, Q-, and advantage functions; goodness-of-fit tests.
result Asymptotic normality of estimators under specific logging policies.
Study random walks on groups with superlinear divergent geodesics.
problem Existence of superlinear divergent geodesics in groups.
method Developed theory of superlinear divergence and applied Gouëzel's pivoting technique.
result Established a central limit theorem for random walks on groups with superlinear divergent geodesics.
Study analyzes a new algorithm for complex optimization problems.
problem Stochastic bilevel optimisation problems in continuous-time models.
method Continuous-time, two-timescale stochastic approximation algorithm.
result Obtained weak convergence rate using central limit theorem.
Paper improves CLT and bootstrap approximations for LSA with decreasing step size.
problem Improving normal approximation and bootstrap methods for LSA with decreasing step sizes.
method Refined Berry-Esseen bounds and multiplier bootstrap procedure for LSA.
result Approximation rates up to 1 / n 1/\sqrt{n} 1/ n for LSA rescaled error distribution. We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on Out ( F N ) \text{Out}(F_N) Out ( F N ) , each time under a finite second moment condition on the measure (either with respect to the Teichmüller metric, or with respect to the Lipschitz metric …
Study improves accuracy of risk measures using advanced algorithms.
problem Computing accurate risk measures for financial losses.
method Nested stochastic approximation and multilevel acceleration.
result Established central limit theorems for estimation errors.
New method for spot volatility estimation with reduced microstructure noise.
problem Estimating spot volatility from noisy high-frequency data.
method Pre-averaging/kernel estimator to handle microstructure noise.
result Optimal bandwidth selection and kernel functions for minimal variance.
The paper strengthens the classical result of MLE convergence to a Gaussian distribution.
problem The classical result of MLE convergence to a Gaussian distribution.
method Sub-Gaussian concentration and entropic normality of the normalized MLE.
result Entropic central limit theorem for a smoothed version of the estimator.
The exchange algorithm is studied for its convergence and asymptotic variance.
problem Theoretical limitations of the exchange algorithm in sampling from doubly-intractable distributions.
method Theoretical analysis of the exchange algorithm's convergence speed and asymptotic variance.
result The exchange algorithm converges at a geometric rate and satisfies a Central Limit Theorem.
The paper connects neural networks to physics using probability theory.
problem Creating neural networks that follow physical laws.
method Applying the central limit theorem and Gaussian process theory to neural networks.
result Neural networks can be designed to obey physical laws by choosing appropriate activation functions.
The paper analyzes Q-learning convergence rates with asynchronous updates.
problem Analyzing convergence rates of asynchronous Q-learning algorithms.
method Derives rates of convergence using high-dimensional central limit theorems.
result Establishes a rate of order up to n − 1 / 6 log 4 ( n S A ) n^{-1/6} \log^{4} (nS A) n − 1/6 log 4 ( n S A ) for hyper-rectangles. Developed moment estimators for affine stochastic volatility models.
problem Estimating parameters of affine stochastic volatility models.
method Introduced recursive equations for moments and proposed moment estimators.
result Established a central limit theorem and derived asymptotic covariance matrix.