arXiv research

Method upgrades limit theorems to mixing limit theorems for dynamical systems.

Paper proves CLTs for Q-learning with asynchronous updates.

The study extends convergence theorems for Ricci-limit spaces with bounded curvature.

Proves limit curve theorem for incomplete metric spaces, applies to null distance in Lorentzian manifolds.

Proves CLT for Brownian paths on pinched negative curvature manifolds.

Central limit theorem for Green metrics on hyperbolic groups.

Stochastic gradient descent's long-term fluctuations are described by a diffusion limit.

In this paper, we compute the adiabatic limit of the scalar curvature and prove several vanishing theorems, we also derive a Kastler-Kalau-Walze type theorem for the noncommutative residue in the case of foliations.

The study of random walks on hyperbolic spaces and Teichmüller spaces, proving central limit theorems and geodesic tracking.

This paper deals with a semi-classical limit (Theorem 1) by using traditional mathematical methods, and shows a Hopf theorem as a corollary. A formal discussion of it may be found in [7].

Paper derives sub-Riemannian versions of Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki theorems for twisted BCV spaces.

Study shows central limit theorem for counting measures in non-smooth spaces.

Survey of recent Kleinian representation convergence results.

Functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting

The study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.

The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.

We consider the task of automated theorem proving, a key AI task. Deep learning has shown promise for training theorem provers, but there are limited human-written theorems and proofs available for supervised learning. To address this limitation, we propose to learn a neural generator that automatically synthesizes the…

We show that the Gromov-Hausdorff limit of a sequence of leaves in a compact foliation is a covering space of the limiting leaf which is no larger than this leaf's holonomy cover. We also show that convergence to such a limit is smooth instead of merely Gromov-Hausdorff. Corollaries include Reeb's local stability theor…

Study shows a central limit theorem for random coverings of manifolds with nilpotent groups.

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…

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…

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…

The study examines Hawkes processes and their long-term behavior.

Study small-time CLTs for stochastic Volterra equations with various kernels.

We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes[Co], whi…

In this paper we prove some general results on constant mean curvature lamination limits of certain sequences of compact surfaces $M_n$ embedded in $\mathbb R^3$ with constant mean curvature $H_n$ and fixed finite genus, when the boundaries of these surfaces tend to infinity. Two of these theorems generalize to the non…

The purpose of this article is to point out a mistake in the published paper "Graphs of hyperbolic groups and limit set intersection theorem- Proc AMS, vol 146, no 5, pp 1859--1871, which subsequently weakens the main theorem of that paper. We state and prove a weaker result in this note.

In this paper we prove a theorem concerning lamination limits of sequences of compact disks $M_n$ embedded in $\mathbb{R}^3$ with constant mean curvature $H_n$, when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi…

Study shows mass distribution of random holomorphic sections follows a central limit theorem.

The authors Balogh-Tyson-Vecchi in arXiv:1604.00180 utilize the Riemannian approximations scheme $(\mathbb H^1,<,>_L)$, in the Heisenberg group, introduced by Gromov, to calculate the limits of Gaussian and normal curvatures defined on surfaces of $\mathbb H^1$ when $L\rightarrow\infty$. They show that these limits exi…

For $α\in (1,2)$, we present a generalized central limit theorem for $α$-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…

The paper calculates curvature limits and Gauss-Bonnet theorems in the Heisenberg group.

We prove a Lorentzian splitting theorem with weakened curvature conditions.

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.

The paper explores properties of functions on Teichmüller space, proving theorems about limits and non-ergodicity.

Study shows superdiffusive behavior in geodesic flows on curved surfaces.

The paper proves rigidity theorems for Type II singularities in Lagrangian flows.

Local limit theorem for random walks on hyperbolic groups with parabolic subgroups.

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 …

A note on utility maximization with costs, proving trading strategies.

In this paper we address the relationship between Gromov-Hausdorff limits and intrinsic flat limits of complete Riemannian manifolds. In \cite{SormaniWenger2010, SormaniWenger2011}, Sormani-Wenger show that for a sequence of Riemannian manifolds with nonnegative Ricci curvature, a uniform upper bound on diameter, and n…

We define a stochastic model of a two-sided limit order book in terms of its key quantities \textit{best bid [ask] price} and the \textit{standing buy [sell] volume density}. For a simple scaling of the discreteness parameters, that keeps the expected volume rate over the considered price interval invariant, we prove a…

Integrability criterion for projective limits of Banach distributions on Fréchet manifolds.

We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel--Mosher and on the outer space of a fr…

Study the topology of Ricci limit spaces using Gromov-Hausdorff limits.

The study proves a quantitative functional CLT for neural networks with smooth activation functions.

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two propertie…

Study k-positive surface group representations and their degenerations.