Finite Moment Research Articles

Local limit theorems without assuming finite third moment

One of the most fundamental probabilities is the probability at a particular point. The local limit theorem is the well-known theorem that estimates this probability. In this paper, we estimate this probability by the density function of normal distribution in the case of lattice integer-valued random variables. Our technique is the characteristic function method. We complete to relax the third moment condition of Siripraparat and Neammanee (J. Inequal. Appl. 2021:57, 2021) and the references therein and also obtain explicit constants of the error bound.

Local existence of weak solutions to kinetic Cucker‐Smale‐Fokker‐Planck equation with singular commutation weights

In this paper, we investigate the Cauchy problem of the dimensional kinetic Cucker‐Smale‐Fokker‐Planck equation with singular communication weight behaving like as . First, local existence of weak solutions with finite kinetic energy is established for and . Second, for , we show that any initial datum with finite velocity moment of order launches a weak solution that propagates the initial velocity moment.

Infinite Volume Gibbs States and Metastates of the Random Field Mean-Field Spherical Model

For the discrete random field Curie–Weiss models, the infinite volume Gibbs states and metastates have been investigated and determined for specific instances of random external fields. In general, there are not many examples in the literature of non-trivial limiting metastates for discrete or continuous spin systems. We analyze the infinite volume Gibbs states of the mean-field spherical model, a model of continuous spins, in a general random external field with independent identically distributed components with finite moments of some order larger than four and non-vanishing variances of the second moments. Depending on the parameters of the model, we show that there exist three distinct phases: ordered ferromagnetic, ordered paramagnetic, and spin glass. In the ordered ferromagnetic and ordered paramagnetic phases, we show that there exists a unique infinite volume Gibbs state almost surely. In the spin glass phase, we show the existence of chaotic size dependence, provide a construction of the Aizenman–Wehr metastate, and consider both the convergence in distribution and almost sure convergence of the Newman–Stein metastates. The limiting metastates are non-trivial and their structure is universal due to the presence of Gaussian fluctuations and the spherical constraint.

Invariance principles for integrated random walks conditioned to stay positive

Let S(n) be a centered random walk with finite second moment. We consider the integrated random walk T(n)=S(0)+S(1)+⋯+S(n). We prove invariance principles for the meander and for the bridge of this process, under the condition that the integrated random walk remains positive. Furthermore, we prove the functional convergence of its Doob’s h-transform to the h-transform of the Kolmogorov diffusion conditioned to stay positive.

Stochastic homogenization of degenerate integral functionals and their Euler-Lagrange equations

We prove stochastic homogenization for integral functionals defined on Sobolev spaces, where the stationary, ergodic integrand satisfies a degenerate growth condition of the form c|ξA(ω,x)| p ≤f(ω,x,ξ)≤|ξA(ω,x)| p +Λ(ω,x) for some p∈(1,+∞) and with a stationary and ergodic diagonal matrix A such that its norm and the norm of its inverse satisfy minimal integrability assumptions and Λ is a nonnegative, stationary function with finite first moment. We also consider the convergence when Dirichlet boundary conditions or an obstacle condition are imposed. Assuming the strict convexity and differentiability of f with respect to its last variable, we further prove that the homogenized integrand is also strictly convex and differentiable. These properties allow us to show homogenization of the associated Euler-Lagrange equations.

Berry–Esseen Bounds with Targets and Local Limit Theorems for Products of Random Matrices

Let $$\mu $$ be a probability measure on $$\textrm{GL}_d(\mathbb {R})$$ and denote by $$S_n:= g_n \cdots g_1$$ the associated random matrix product, where $$g_j$$ ’s are i.i.d.’s with law $$\mu $$ . We study statistical properties of random variables of the form $$\begin{aligned} \sigma (S_n,x) + u(S_n x), \end{aligned}$$ where $$x \in \mathbb {P}^{d-1}$$ , $$\sigma $$ is the norm cocycle and u belongs to a class of admissible functions on $$\mathbb {P}^{d-1}$$ with values in $$\mathbb {R}\cup \{\pm \infty \}$$ . Assuming that $$\mu $$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry–Esseen bounds and the Local Limit Theorem for such variables using a large class of observables on $$\mathbb {R}$$ and Hölder continuous target functions on $$\mathbb {P}^{d-1}$$ . As particular cases, we obtain new limit theorems for $$\sigma (S_n,x)$$ and for the coefficients of $$S_n$$ .

Ensemble approximations for constrained dynamical systems using Liouville equation

For a class of state-constrained dynamical systems described by evolution variational inequalities, we study the time evolution of a probability measure which describes the distribution of the state over a set. In contrast to smooth ordinary differential equations, where the evolution of this probability measure is described by the Liouville equations, the flow map associated with the nonsmooth differential inclusion is not necessarily invertible and one cannot directly derive a continuity equation to describe the evolution of the distribution of states. Instead, we consider Lipschitz approximation of our original nonsmooth system and construct a sequence of measures obtained from Liouville equations corresponding to these approximations. This sequence of measures converges in weak-star topology to the measure describing the evolution of the distribution of states for the original nonsmooth system. This allows us to approximate numerically the evolution of moments (up to some finite order) for our original nonsmooth system, using a solver that uses finite order moment approximations of the Liouville equation. Our approach is illustrated with the help of two academic examples.

Local mass-conserving solution for a critical coagulation-fragmentation equation

The critical coagulation-fragmentation equation with multiplicative coagulation and constant fragmentation kernels is known to not have global mass-conserving solutions when the initial mass is greater than 1. We show that for any given positive initial mass with finite second moment, there is a time T⁎>0 such that the equation possesses a unique mass-conserving solution up to T⁎. The novel idea is to singularly perturb the constant fragmentation kernel by small additive terms and study the limiting behavior of the solutions of the perturbed system via the Bernstein transform.

The stable graph: The metric space scaling limit of a critical random graph with i.i.d. power-law degrees

We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent α+1, where α∈(1,2). The limiting components are constructed from random R-trees encoded by the excursions above its running infimum of a process whose law is locally absolutely continuous with respect to that of a spectrally positive α-stable Lévy process. These spanning R-trees are measure-changed α-stable trees. In each such R-tree, we make a random number of vertex identifications, whose locations are determined by an auxiliary Poisson process. This generalises results, which were already known in the case where the degree distribution has a finite third moment (a model which lies in the same universality class as the Erdős–Rényi random graph) and where the role of the α-stable Lévy process is played by a Brownian motion.

A Controlled Mean Field Model for Chiplet Population Dynamics

In micro-assembly applications, ensemble of chiplets immersed in a dielectric fluid are steered using dielectrophoretic forces induced by an array of electrode population. Generalizing the finite population deterministic models proposed in prior works for individual chiplet position dynamics, we derive a controlled mean field model for a continuum of chiplet population in the form of a nonlocal, nonlinear partial differential equation. The proposed model accounts for the stochastic forces as well as two different types of nonlocal interactions, viz. chiplet-to-chiplet and chiplet-to-electrode interactions. Both of these interactions are nonlinear functions of the electrode voltage input. We prove that the deduced mean field evolution can be expressed as the Wasserstein gradient flow of a Lyapunov-like energy functional. With respect to this functional, the resulting dynamics is a gradient descent on the manifold of joint population density functions with finite second moments that are supported on the position coordinates.

Some asymptotic properties of random walks on homogeneous spaces

Let $ G $ be a connected semisimple real Lie group with finite center, and $ \mu $ a probability measure on $ G $ whose support generates a Zariski-dense subgroup of $ G $. We consider the right $ \mu $-random walk on $ G $ and show that each random trajectory spends most of its time at bounded distance of a well-chosen Weyl chamber. We infer that if $ G $ has rank one, and $ \mu $ has a finite first moment, then for any discrete subgroup $ \Lambda\subseteq G $, the $ \mu $-walk and the geodesic flow on $ \Lambda \backslash G $ are either both transient, or both recurrent and ergodic, thus extending a well known theorem due to Hopf–Tsuji–Sullivan–Kaimanovich dealing with the Brownian motion.

Anomalous diffusion, gigahertz, and terahertz spectroscopy of aliphatic ketones.

The experimental results of the complex dielectric permittivity of aliphatic ketones in dilute solutions of inert solvent cyclohexane in the gigahertz (GHz) and terahertz (THz) frequencies of the electromagnetic spectrum are examined in terms of the theory of inertial anomalous diffusion of polar molecules, considered as an assembly of molecules with interacting dipolar groups, in polar liquids. The theory is based on the generalization of the Debye rotational diffusion model of dielectric relaxation of polar molecules. The model comprises two interacting dipolar groups-one lighter and the other heavier; each has a finite moment of inertia and each experiences a finite friction with an extensive range of damping or drag coefficient and the dipole moment ratio of the two groups. The lighter group refers to the reference molecule, whereas the heavier group simulates the neighboring molecules, and the two groups interact with each other via the dipole-dipole interaction potential. The resulting approximate expression contains terms in both the Rocard form and the Сole-Сole form. The experimental data on aliphatic ketones are shown to fit extremely well with the theory, and parameters of the fit offer physical significance. An agreement of the plot of the experimental data on dielectric loss versus frequency to the formulas derived from the model offers a mathematical basis of the semiempirical equations used in the literature to fit the experimental data. Experimental results of the dielectric loss of neat polar liquid acetone in GHz and THz, as an example, are shown to fit the theory; however, the interaction potential parameter compared to its dilute solution counterpart is significantly increased, reflecting the increase in the dipole-dipole interaction energy.

Multivariate [formula omitted]-stable distributions: VAR(1) processes, measures of dependence and their estimations

Measuring the dependency between lagged random vectors in time series is extremely important to detect appropriate models for fitting real data. Vector autoregressive processes (VAR), in particular, are among the most popular multivariate time series that allow modeling schemes for two or more random variables. However, if the series of innovations process does not have finite second-order moments, the classical analysis that uses the empirical auto-covariance function is not adequate as its theoretical counterpart is not defined. After perceiving the lack of a suitable multivariate version of the codifference function, we propose an adaptation of this tool to measure interdependence in multivariate α-stable processes. We establish such a function based on notions presented in the existing literature. We also show some of its properties and give an estimator based on the empirical characteristic function. Subsequently, we evaluate another measure of dependence with different properties but similar usage for comparison reasons. These two measures are computed for VAR(1) processes with α-stable innovations, proving to be useful in the identification of associated patterns related to them. Finally, we demonstrate how these measures can be applied with real data sets by analyzing the monthly number of patients with community-acquired pneumonia and the registered values of PM10 in the city of São Paulo, Brazil, from January 2008 to December 2021.

BERRY–ESSEEN BOUND AND LOCAL LIMIT THEOREM FOR THE COEFFICIENTS OF PRODUCTS OF RANDOM MATRICES

AbstractLet $\mu $ be a probability measure on $\mathrm {GL}_d(\mathbb {R})$ , and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$ are i.i.d. with law $\mu $ . Under the assumptions that $\mu $ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we prove a Berry–Esseen bound with the optimal rate $O(1/\sqrt n)$ for the coefficients of $S_n$ , settling a long-standing question considered since the fundamental work of Guivarc’h and Raugi. The local limit theorem for the coefficients is also obtained, complementing a recent partial result of Grama, Quint and Xiao.

Equidistribution of mass for random processes on finite-volume spaces

Let G be a real Lie group, Λ ⊆ G a lattice, and X = G/Λ. We fix a probability measure μ on G and consider the left random walk induced on X. It is assumed that μ is aperiodic, has a finite first moment, spans a semisimple algebraic group without compact factors, and has two non mutually singular convolution powers. We show that for every starting point x ∈ X, the n-th step distribution μn*δx of the walk weak-* converges toward some homogeneous probability measure on X.

Self-similar Profiles for Homoenergetic Solutions of the Boltzmann Equation for Non-cutoff Maxwell Molecules

We consider a modified Boltzmann equation which contains, together with the collision operator, an additional drift term which is characterized by a matrix A. Furthermore, we consider a Maxwell gas, where the collision kernel has an angular singularity. Such an equation is used in the study of homoenergetic solutions to the Boltzmann equation. Under smallness assumptions on the drift term, we prove that the longtime asymptotics is given by self-similar solutions. We work in the framework of measure-valued solutions with finite moments of order p>2 and show existence, uniqueness and stability of these self-similar solutions for sufficiently small A. Furthermore, we prove that they have finite moments of arbitrary order if A is small enough. In addition, the singular collision operator allows to prove smoothness of these self-similar solutions. Finally, we study the asymptotics of particular homoenergetic solutions. This extends previous results from the cutoff case to non-cutoff Maxwell gases.

Asymptotic optimality of the binomial-exhaustive policy for polling systems with large switchover times

We study an optimal-control problem of polling systems with large switchover times, when a holding cost is incurred on the queues. In particular, we consider a stochastic network with a single server that switches between several buffers (queues) according to a pre-specified order, assuming that the switchover times between the queues are large relative to the processing times of individual jobs. Due to its complexity, computing an optimal control for such a system is prohibitive, and so we instead search for an asymptotically optimal control. To this end, we first solve an optimal control problem for a deterministic relaxation (namely, for a fluid model), that is represented as a hybrid dynamical system. We then “translate” the solution to that fluid problem to a binomial-exhaustive policy for the underlying stochastic system, and prove that this policy is asymptotically optimal in a large-switchover-time scaling regime, provided a certain uniform integrability (UI) condition holds. Finally, we demonstrate that the aforementioned UI condition holds in the following cases: (i) the holding cost has (at most) linear growth, and all service times have finite second moments; (ii) the holding cost grows at most at a polynomial rate (of any degree), and the service-time distributions possess finite moment generating functions.

On Estimating Rank-One Spiked Tensors in the Presence of Heavy Tailed Errors

In this paper, we study the estimation of a rank-one spiked tensor in the presence of heavy tailed noise. Our results highlight some of the fundamental similarities and differences in the tradeoff between statistical and computational efficiencies under heavy tailed and Gaussian noise. In particular, we show that, for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> th order tensors, the tradeoff manifests in an identical fashion as the Gaussian case when the noise has finite <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4(p-1)$ </tex-math></inline-formula> th moment. The difference in signal strength requirements, with or without computational constraints, for us to estimate the singular vectors at the optimal rate, interestingly, narrows for noise with heavier tails and vanishes when the noise only has finite fourth moment. Moreover, if the noise has less than fourth moment, tensor SVD, perhaps the most natural approach, is suboptimal even though it is computationally intractable. Our analysis exploits a close connection between estimating the rank-one spikes and the spectral norm of a random tensor with iid entries. In particular, we show that the order of the spectral norm of a random tensor can be precisely characterized by the moment of its entries, generalizing classical results for random matrices. In addition to the theoretical guarantees, we propose estimation procedures for the heavy tailed regime, which are easy to implement and efficient to run. Numerical experiments are presented to demonstrate their practical merits.

A NEGATIVE BINOMIAL AUTOREGRESSION WITH A LINEAR CONDITIONAL VARIANCE-TO-MEAN FUNCTION

A general integer-valued time-series model with a conditional variance proportional to the conditional mean is proposed. Specifically, the conditional distribution is a Poisson mixture with a dependent mixing sequence, which results in a negative binomial distribution with a linear conditional variance-to-mean relationship. In addition, the conditional mean is specified as a general parametric function of past observations. We first propose stationarity, ergodicity, and finite moment conditions for the model. Furthermore, the parameters are estimated using the Poisson quasi-maximum likelihood estimate, whose asymptotic properties are studied under weak conditions. Illustrations of the proposed methodology on simulated and actual time series of counts are given.

Stochastic homogenization: a short proof of the annealed Calderón–Zygmund estimate

A building block for many field theories in continuum physics are second-order elliptic operators in divergence form, as given through a coefficient field which may be assimilated to a metric tensor field on \(\mathbb {R}^d\). The mapping properties of these linear operators are a crucial ingredient for analysis. In this paper, we focus on Calderón–Zygmund estimates, that is, on the boundedness of the corresponding Helmholtz projection in \(\mathrm{{L}}^p(\mathbb {R}^d)\)-spaces. Even when the coefficient field is uniformly smooth, this estimate may fail for p not close to 2. We seek an intrinsic criterion on the validity of the Calderón–Zygmund estimate in the whole range of \(p\in (1,\infty )\); intrinsic in the sense that it is formulated in terms of the scalar and vector potentials of the harmonic coordinates. We seek genericity in form of a statistical statement, and thus consider general ensembles of coefficient fields. Our criterion comes in form of finite stochastic moments for the potentials, or rather their corrections from being affine. In line with this, the Calderón–Zygmund estimates we obtain are annealed as opposed to quenched, meaning that there is an inner norm in form of a stochastic moment next to the (outer) \(\mathrm{{L}}^p\)-norm in space. This result grows out of recent progress in quantitative stochastic homogenization; it is ultimately inspired by the classical large-scale regularity theory of Avellaneda and Lin (Commun Pure Appl Math 40(6):803–847, 1987). More specifically, we provide an easier version of the proof given by us in Josien and Otto (J Funct Anal, 2022), albeit under stronger assumptions. Annealed Calderon–Zygmund estimates were first established in Duerinckx and Otto (Stoch Partial Differ Equ Anal Comput 8(3):625–692, 2020), and are a very convenient tool for error estimates in stochastic homogenization.