The paper discusses the importance of infinite-mean models in finance and risk management.
problem Classic statistical models assume finite mean or variance, which is not suitable for heavy-tailed data.
method Discussion and recent results on infinite-mean models in economics and finance.
result Classic statistical results for finite-mean models often fail or flip for infinite-mean models.
One of the goals of this article is to define a an unified setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure. We first remark that some known examples coming from the theory of metric measured spaces and also from oscill…
Insurance benefits risk sharing for finite mean risks but not for infinite mean risks.
problem The effect of risk sharing and diversification for infinite mean risks.
method Investigation of risk sharing and diversification for infinite mean models, including stable, Pareto, and Fréchet distributions.
result Risk sharing can have a negative effect for infinite mean models, a phenomenon known as the nondiversification trap.
The paper constructs infinitely many surfaces with specific mean curvature.
problem Creating surfaces with prescribed mean curvature in the presence of a strictly stable minimal surface.
method Synthesizing ideas from previous constructions to create multiple surfaces.
result Infinitely many distinct surfaces with prescribed mean curvature are constructed.
We study a moduli space of ASD connections over S3×R. We consider not only finite energy ASD connections but also infinite energy ones. So the moduli space is infinite dimensional in general. We study the (local) mean dimension of this infinite dimensional moduli space. We show the upper bound on the …
New findings allow infinite mean intensity Hawkes processes to be stable.
problem Stability condition for Hawkes processes with infinite mean intensity.
method Analysis of Quadratic Hawkes processes with infinite mean intensity.
result Quadratic Hawkes processes are always stationary with infinite mean intensity when total endogeneity ratio exceeds unity.
Study examines infinite limits of transformer dynamics, identifying key parameterizations.
problem Understanding the training dynamics of transformer models in the feature learning regime.
method Analysis of infinite scaling limits using dynamical mean field theory.
result Identified parameterizations that admit well-defined infinite width and depth limits.
The paper constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.
problem Infinite-time singularities in Lagrangian mean curvature flow.
method Constructing solutions by gluing special Lagrangian 'Lawlor necks' and analyzing dynamics of neck size.
result The flow decomposes initial data into a union of special Lagrangians intersecting at one point.
Paper establishes sufficient condition for comparing linear combinations of infinite-mean risks.
problem Comparing linear combinations of infinite-mean risks under stochastic dominance.
method Introduced a new class of distributions and used majorization order to compare weights.
result Linear combinations of random variables are stochastically larger when their weight vectors are smaller in majorization order.
Bayesian neural networks learn efficiently at infinite width, matching polynomial-width performance.
problem Understanding the inductive bias of infinite-width neural networks.
method Analyzing the reduced entropy and using subsampling techniques.
result The Bayesian mean-field learner generalizes exactly on polynomially-bounded targets.
The study finds either many or few constant mean curvature hypersurfaces on a manifold.
problem Finding constant mean curvature hypersurfaces on a manifold.
method Analyzing a manifold with a generic Riemannian metric to determine the existence of hypersurfaces.
result Either infinitely many or infinitely many hypersurfaces with specific mean curvatures exist.
This paper studies the Yang--Mills ASD equation over the cylinder as a non-linear evolution equation. We consider a dynamical system consisting of bounded orbits of this evolution equation. This system contains many chaotic orbits, and moreover it becomes an infinite dimensional and infinite entropy system. We study th…
Unified asymptotic theory and tests for ACD models reveal infinite-mean durations in cryptocurrency trading.
problem Challenges in asymptotic theory for ACD models, especially for integrated ACD.
method Unified asymptotic theory for quasi-maximum likelihood estimator, hypothesis testing framework.
result Infinite-mean durations in cryptocurrency trading, rejected integrated ACD hypothesis.
The paper constructs singularities for Lagrangian flow in Gibbons-Hawking spaces with vanishing mean curvature.
problem Infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons-Hawking spaces.
method One-parameter family of barrier curves and detailed asymptotic analysis.
result The mean curvature converges uniformly to zero, but the second fundamental form becomes unbounded.
Study of LQ MFGs in infinite-dimensional Hilbert spaces.
problem Mean field games in infinite-dimensional settings with stochastic dynamics.
method Analysis of coupled semilinear infinite-dimensional stochastic evolution equations, development of Nash equilibrium.
result Characterization of unique Nash equilibrium in the limit of many agents.
New calibration energy measures deviation from calibrated geometry, enabling mean curvature flow in infinite volumes.
problem Mean curvature flow in infinite volumes with finite energy.
method Introducing calibration energy and proving its dissipation identity for mean curvature flows.
result Every proper self-expander with finite calibration energy is a plane in all dimensions and codimensions.
Paper controls shape stability in infinite Riemannian manifolds.
problem Characterizing optimal shapes in infinite-dimensional Riemannian manifolds.
method Uses Riemannian manifold framework and mean curvature analysis.
result Control on shape stability depends only on mean curvature.
We study an infinite dimensional ASD moduli space over the cylinder. Our main result is the formula of its local mean dimension. A key ingredient of the argument is the notion of non-degenerate ASD connections. We develop its deformation theory and show that there exist sufficiently many non-degenerate ASD connections …
Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
problem Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
method Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
result Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
Study finds infinitely many non-radial solutions for negative scalar curvature in higher dimensions.
problem Prescribing scalar and boundary mean curvature in a ball with negative scalar curvature.
method Analyzes the existence of infinitely many non-radial positive solutions in dimensions 5 and above.
result First existence result for negative scalar curvature in higher dimensions.
Developed LQ MFG theory with common noise, proving existence and uniqueness.
problem Linear-quadratic mean field games with common noise.
method Coupled forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces.
result Existence and uniqueness of solutions for small and arbitrary finite time horizons.
This paper presents a natural extension of stagewise ranking to the the case of infinitely many items. We introduce the infinite generalized Mallows model (IGM), describe its properties and give procedures to estimate it from data. For estimation of multimodal distributions we introduce the Exponential-Blurring-Mean-Sh…
The study finds many Möbius bands and annuli on toroids.
problem Finding minimal surfaces on toroids.
method Equivariant variational methods.
result Linear growth of surface areas with symmetry order.
New framework for understanding infinite-width neural networks.
problem Understanding the infinite-width limit behavior of neural networks.
method General framework to study limit behavior of neural models based on hyperparameter scaling.
result Derives scaling for existing mean-field and neural tangent kernel limits and introduces new dynamically stable limits.
We prove that rational homology of the Torelli group of genus g is infinite dimensional, provided g>6. This means that rational homology of the Torelli space of genus g>6 is infinite dimensional. The Torelli groups with marked points are also considered. In addition, we prove that rational homology of the subgroup of t…
Paper extends stochastic dominance for compound binomial distributions.
problem Stochastic dominance for infinite-mean random variables.
method Investigates properties and inclusion relationships of distribution classes, extends results to compound binomial distributions.
result Establishes necessary and sufficient conditions for first-order stochastic dominance preservation.
The paper shows how Scherk-type surfaces can be decomposed into helicoids.
problem Decomposing Scherk-type zero mean curvature surfaces.
method Using a special Euler-Ramanujan identity and Wick rotation, the paper expresses these surfaces as an infinite superposition of dilated helicoids and provides different finite decompositions.
result Scherk-type zero mean curvature surfaces can be expressed as an infinite superposition of dilated helicoids.
The main purpose of this paper is to show that ideas of deformation theory can be applied to "infinite dimensional geometry". We develop the deformation theory of Brody curves. Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter …
Study shows polynomial-width neural networks can closely approximate infinite-width networks in polynomial time.
problem Approximating dynamics of polynomial-width neural networks with infinite-width networks.
method Bounding approximation gap through a differential equation governed by mean-field dynamics, considering local Hessian.
result Polynomially many neurons are sufficient to closely approximate mean-field dynamics.
In this paper we construct an end of a self-similar shrinking solution of the mean curvature flow asymptotic to an isoparametric cone C and lying outside of C. We call a cone C in Rn+1 an isoparametric cone if C is the cone over a compact embedded isoparametric hypersurface Γ⊂Sn. The theory of isoparamet…
Unexpectedly, weighted Pareto variables are stochastically dominant.
problem Understanding stochastic dominance in Pareto distributions.
method Analyzing weighted averages of Pareto random variables with infinite mean.
result The weighted average of Pareto variables is stochastically dominant.
The paper finds infinitely many half-volume CMC hypersurfaces on generic or Ricci-positive manifolds.
problem Finding infinitely many half-volume constant mean curvature (CMC) hypersurfaces on manifolds.
method Developed a min-max theory for non-local functionals to prove the existence of these hypersurfaces.
result Infinitely many geometrically distinct CMC hypersurfaces enclosing half the volume of a manifold.
We connect shift-invariant characteristic kernels to infinitely divisible distributions on Rd. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures. The contribution of this paper is two-fold. First, we show, usi…
A new algorithm for learning shallow neural networks with infinite width.
problem Learning shallow over-parameterized neural networks.
method Sinkhorn proximal algorithm approximating mean field learning dynamics.
result The algorithm performs gradient descent of the free energy associated with the risk functional.
Bayesian neural networks ignore data in infinite units limit.
problem Pathological behavior of posterior in over-parameterized networks.
method Mean-field variational inference in infinite hidden units limit.
result Posterior mean converges to zero, ignoring data.
We use bifurcation theory to show the existence of infinite sequences isometric embeddings of tori with constant mean curvature (CMC) in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus.
The paper tackles a bandit problem with infinitely many arms per group, aiming to identify the group with the highest quantile reward.
problem Max-quantile group bandit problem with infinitely many arms per group.
method Two-step algorithm: first request arms from each group, then apply a finite-arm max-quantile bandit algorithm.
result Characterization of instance-dependent and worst-case regret, with matching lower bounds.
Infinite CNNs lose spatial correlations, but can be restored by correlated weights.
problem Infinite CNNs lose spatial correlations, which are crucial for their performance.
method Introduced correlated weights to restore spatial correlations in infinite CNNs.
result Optimal performance is achieved with a moderate level of weight correlation.
In this paper, we consider infinite-dimensional port-Hamiltonian systems with in-domain actuation by means of an approach based on Stokes-Dirac structures as well as in a framework that exploits an underlying jet-bundle structure. In both frameworks, a dynamic controller based on the energy-Casimir method is derived in…
Neural operators learn to solve LQ MFGs efficiently in infinite dimensions.
problem Solving many related LQ MFG problems in infinite-dimensional settings.
method Training neural operators to map problem data to equilibrium strategies.
result NOs reliably solve unseen LQ MFG variants with controlled parameters.
New 4-manifolds with exotic diffeomorphisms found.
problem Existence of exotic diffeomorphisms in 4-manifolds.
method Proves existence of infinitely many contractible 4-manifolds with exotic diffeomorphisms.
result Infinitely many contractible 4-manifolds with absolutely exotic diffeomorphisms.
In this paper we find functions over bounded domains in the 2-dimensional Euclidean space, whose graphs (in the Heisenberg space) has constant mean curvature different from zero and taking on (possibly) infinite boundary values over the boundary of the domain.
Study shows submanifolds can't be immersed in certain spaces.
problem Non-immersibility of submanifolds with infinite mean exit time.
method Not based on the weak maximum principle at infinity, generalizes previous results.
result Estimates for complete tower of moments for submanifolds with small mean curvature.
Study proves mean curvature flows on spheres in higher dimensions.
problem Existence of mean curvature flows on spheres.
method Generalized previous results to higher dimensions, proving existence of flows.
result Existence of infinitely many eternal weak mean curvature flows in Sn+1 connecting specific hypersurfaces. We construct an infinite order cork (W,f), which means that W is a smooth compact contractible 4-manifold with Stein structure, and f is a self diffeomorphism of the boundary of W, such that the n-fold composition maps f^{n}=f o f o... o f give rise to smoothly distinct corks (W, f^{n}) for sufficiently large values of…
This paper shows how infinitely wide Tensor Networks converge to Gaussian Processes.
problem Understanding the relationship between Tensor Networks and Gaussian Processes.
method Analyzing the infinite-width limit of Tensor Networks and comparing them to Gaussian Processes.
result Infinitely wide Tensor Networks converge to Gaussian Processes, proving their equivalence.
Diversification improves profits for heavy-tailed investments.
problem Investment portfolios of Pareto-distributed returns.
method Stochastic dominance and majorization order.
result Diversification increases first-order stochastic dominance for heavy-tailed returns.
Unified meta algorithms estimate various distribution functionals in infinite-armed bandits.
problem Estimating various distribution functionals in infinite-armed bandits.
method Unified meta algorithms for offline and online settings, achieving optimal sample complexities.
result Online estimation offers significant advantage for certain distribution functionals.