Finite Moment Research Articles

Truthful and privacy-preserving generalized linear models

This paper explores estimating Generalized Linear Models (GLMs) when agents are strategic and privacy-conscious. We aim to design mechanisms that encourage truthful reporting, protect privacy, and ensure outputs are close to the true parameters. Initially, we address models with sub-Gaussian covariates and heavy-tailed responses with finite fourth moments, proposing a novel private, closed-form estimator. Our mechanism features: (1) o(1)-joint differential privacy with high probability; (2) o(1n)-approximate Bayes Nash equilibrium for (1−o(1))-fraction of agents; (3) o(1) error in parameter estimation; (4) individual rationality for (1−o(1)) of agents; (5) o(1) payment budget. We then extend our approach to linear regression with heavy-tailed data, using an ℓ4-norm shrinkage operator to propose a similar estimator and payment scheme.

Electronic and Magnetic Properties of Cobalt Clusters on Pristine and Divacancy Graphene

Using ab initio calculations we investigate the adsorption of Co atoms, dimers and small cobalt clusters of 5 and 13 atoms on pristine graphene and graphene with a double vacancy. We report the atomic, electronic, magnetic and energetic properties of these systems. Stable adsorption configurations tend to maximize the number of cobalt-carbon bonds. On graphene, the adsorption energy of the clusters is only about 0.4 to 1 eV, and the clusters are relatively mobile on graphene. Interestingly, for different adsorbed Co13 isomers on graphene it is found that they converge to the same atomic structure. On graphene with a divacancy, the Co clusters bind in the divacancy site and isomerisation also occurs for the Co5 cluster system as well as for Co13. Co atoms and clusters can be effectively immobilized on the divacancy with corresponding adsorption energy being significantly enhanced by about 5 to 7 eV. All clusters act as electron donors in the interaction with the graphene/divacancy systems, and the amount of electron charge transfer increases with cluster size. Finite magnetic moments occur for all systems, where upon adsorption, the magnetic moment of the isolated Co atom (3μB) is significantly reduced due to electron transfer and bonding, resulting in values varying from ≈0.9-2.2 μB per Co atom. For the pristine graphene substrate, the total induced magnetic moments on the carbon atoms are negligible, while on the divacancy system, they are of the order of 0.1-0.3 μB. The attractive physical properties of these hybrid systems could find applications in catalysis and materials science.

Strong convergence of the empirical spectral distribution of a unified matrix model

Let [Formula: see text] is a positive definite matrix and [Formula: see text] with [Formula: see text] being independent and identically distributed (i.i.d.) centered real random variables. Consider the matrix [Formula: see text] where [Formula: see text] and [Formula: see text] are two projection matrices (deterministic or random) satisfying [Formula: see text] and [Formula: see text]. Additionally, if [Formula: see text] and [Formula: see text] are random matrices, they are independent of [Formula: see text]. In this paper, we demonstrate that the empirical spectral distribution of [Formula: see text] converges almost surely to a non-random distribution when [Formula: see text] and [Formula: see text], assuming [Formula: see text] has finite second moment.

A Markovian Gauss inequality for asymmetric deviations from the mode of symmetric unimodal distributions

For a random variable with a unimodal distribution and finite second moment Gauß (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it under the assumption of a finite r r -th absolute moment is given ( r ≥ 1 r\geq 1 ), based on the Khintchine representation of unimodal random variables. A special instance of the resulting Narumi–Gauß inequality is the one with finite first absolute moment, which might be called a Markovian Gauß inequality. For symmetric unimodal distributions with finite second moment Semenikhin (2019) generalized the Gauß inequality to arbitrary intervals. For the class of symmetric unimodal distributions with finite first absolute moment we construct a Markovian version of it. Related inequalities of Volkov (1969) and Sellke and Sellke (1997) will be discussed as well.

When Heavy Tails Disrupt Statistical Inference

Heavy tails (HT) arise in many applications and their presence can disrupt statistical inference, yet the HT statistical literature requires a theoretical background most practicing statisticians lack. We provide an overview of the influence of HT on the performance of basic statistical methods and useful theorems aimed at the practitioner encountering HT in an applied setting. Higher or even lower product moments (i.e., variance, skewness, etc.) can be infinite for some HT populations, yet all L-moments are always finite, given that the mean exists, thus, the theory of L-moments is uniquely suited to all HT distributions and data. We document how L-kurtosis, (a kurtosis measure based on the fourth L-moment) provides a general and practical heaviness index for contrasting tail heaviness across distributions and datasets and how a single L-moment diagram can document both the prevalence and impact of HT distributions and data across disciplines and datasets. Surprisingly, the theory of L-moments, an extension and evolution of probability weighted moments, has been largely overlooked by the literature on HT distributions that exhibit infinite moments. Experiments reveal L-kurtosis ranges under which various HT distributions result in mild to severe disruption to the bootstrap, the central limit theorem (CLT), and the law of large numbers, even for distributions which exhibit finite product moments.

The Poisson boundary of lampshuffler groups

AbstractWe study random walks on the lampshuffler group $$\textrm{FSym}(H)\rtimes H$$ FSym ( H ) ⋊ H , where H is a finitely generated group and $$\textrm{FSym}(H)$$ FSym ( H ) is the group of finitary permutations of H. We show that for any step distribution $$\mu $$ μ with a finite first moment that induces a transient random walk on H, the permutation coordinate of the random walk almost surely stabilizes pointwise. Our main result states that for $$H=\mathbb {Z}$$ H = Z , the above convergence completely describes the Poisson boundary of the random walk $$(\textrm{FSym}(\mathbb {Z})\rtimes \mathbb {Z},\mu )$$ ( FSym ( Z ) ⋊ Z , μ ) .

Convergence of unforced errors in the matching problem

The Central Limit Theorem (CLT) is a fundamental result in probability theory with numerous applications in various disciplines. There are a wide variety of CLT‐like theorems, for example the De Moivre–Laplace, Lindeberg and Lyapounov theorems. Independent and identically distributed random variables with a finite second moment are usually considered assumptions in CLTs. However, in this article we present a CLT for a sum of dependent and nonidentically distributed Bernoulli random variables that arises in the well‐known matching experiment.

Exponential and Pareto-type bounds for the renewal function and the excess lifetime of a renewal process

In this paper, our objective is to derive two-sided bounds for the renewal function even if the interarrival times have no finite moment. By proving a relationship between the renewal function and a suitable ‘truncated' compound geometric distribution, under a truncated generalized Lundberg condition for proper renewal equations, we derive upper and lower bounds for this truncated compound geometric distribution. Using this relationship, we obtain bounds for the renewal function. Under specific truncated generalized Lundberg conditions, we obtain two-sided exponential-type bounds and Pareto-type lower bounds for the renewal function. Also, by proving a relationship between the excess lifetime of a renewal process and the df of a suitable ‘truncated deficit at ruin' of a renewal risk (or Sparre Andersen) model, we obtain two-sided exponential bounds for the df of the excess lifetime.

Large sample correlation matrices with unbounded spectrum

In this paper, we demonstrate that the diagonal of a high-dimensional sample covariance matrix stemming from n independent observations of a p-dimensional time series with finite fourth moments can be approximated in spectral norm by the diagonal of the population covariance matrix regardless of the spectral norm of the population covariance matrix. Our assumptions involve p and n tending to infinity, with p/n tending to a constant which might be positive or zero. Consequently, we investigate the asymptotic properties of the sample correlation matrix with a divergent spectrum, and we explore its applications by deriving the limiting spectral distribution for its eigenvalues and analyzing the convergence of divergent and non-divergent spiked eigenvalues under a generalized spiked correlation framework.

Bulk universality for complex non-Hermitian matrices with independent and identically distributed entries

AbstractWe consider $$N\times N$$ N × N matrices with complex entries that are perturbed by a complex Gaussian matrix with small variance. We prove that if the unperturbed matrix satisfies certain local laws then the bulk correlation functions are universal in the large N limit. Assuming the entries are independent and identically distributed (iid) with a common distribution that has finite moments, the Gaussian component is removed by the four moment theorem of Tao and Vu.

Ni thin-films on Pd surfaces and effects of oxygen adsorption: Ab-initio study of structures, electronic properties, magnetic anisotropy

We report first-principles electronic structure calculations of the structural, electronic, and magnetic properties of model epitaxial layers consisting of nickel (Ni) atomic layers deposited on palladium (Pd) substrate, i.e., Ni(001)m∣Pd(001)n where m=1,2,6 and n=3,10, are layer thicknesses. We also investigate the effect of oxygen adsorption on the calculated properties. We found variation in magnetization of between ≈0.6μB to 1.00 μB across the nickel layers. Also, finite magnetic moments albeit of small values of between 0.2 μB and 0.3 μB is found on the Pd at the interface. This magnetic moment on an otherwise non-magnetic Pd atoms has been adduced to interfacial strain due to lattice mismatch between the Ni and Pd layers at the Ni|Pd interface. The effect of adsorbed oxygen on the Nim∣Pdn is that it increases the magnetic moment on the nickel layers. Also, regarding the magnitude of magnetic anisotropy energy (MAE), we found a high perpendicular values of 1.63 meV and 1.37 meV per unit cell respectively for Nim∣Pd10 (m=2,6) which are relatively higher than those reported for other transition metal epitaxial layers. However, the presence of oxygen atom on the Ni∣Pd changes the direction and magnitude of MAE. Indeed, O adsorption favours or enhances in-plane magnetization direction depending on the thickness of the Ni layers for a fixed Pd thickness. Plots of local density of states (LDOS) which include the effect of spin–orbit coupling (SOC), show that in the case of Ni∣Pd having perpendicular MAE, there appears a new SOC-induced electronic states below and above the Fermi level. These states appears to stabilize this type of magnetic anisotropy. On the other hand, in-plane MAE is characterized by SOC-induced localized states below the Fermi level (EF) as well as the lowering of the DOS at the EF. Our work explores the physical, magnetic and electronic properties that may be useful in designing Ni∣Pd-based superlattices for magnetic or spintronic applications.

Central limit theorem in disordered Monomer‐Dimer model

AbstractWe consider the disordered monomer‐dimer model on general finite graphs with bounded degrees. Under the finite fourth moment assumption on the weight distributions, we prove a Gaussian central limit theorem for the free energy of the associated Gibbs measure with a rate of convergence. The central limit theorem continues to hold under a nearly optimal finite ‐moment assumption on the weight distributions if the underlying graphs are further assumed to have a uniformly subexponential volume growth. This generalizes a recent result by Dey and Krishnan who showed a Gaussian central limit theorem in the disordered monomer‐dimer model on cylinder graphs. Our proof relies on the idea that the disordered monomer‐dimer model exhibits a decay of correlation with high probability. We also establish a central limit theorem for the Gibbs average of the number of dimers where the underlying graph has subexponential volume growth and the edge weights are Gaussians.

Origin of Magnetism in a Supposedly Nonmagnetic Osmium Oxide.

A supposedly nonmagnetic 5d^{1} double perovskite oxide is investigated by a combination of spectroscopic and theoretical methods, namely, resonant inelastic x-ray scattering, x-ray absorption spectroscopy, magnetic circular dichroism, and multiplet ligand-field calculations. We found that the large spin-orbit coupling admixes the 5d t_{2g} and e_{g} orbitals, covalency raises the 5d population well above the nominal value, and the local symmetry is lower than O_{h}. The obtained electronic interactions account for the finite magnetic moment of Os in this compound and, in general, of 5d^{1} ions. Our results provide direct evidence of elusive Jahn-Teller distortions, hinting at a strong electron-lattice coupling.

Screening and Antiscreening Effects in Endohedral Nanotubes.

Recently we investigated from first-principles screening properties in systems where small molecules, characterized by a finite electronic dipole moment, are encapsulated in different nanocages. The most relevant result was the observation of an antiscreening effect in alkali-halide nanocages characterized by ionic bonds: in fact, due to the relative displacement of positive and negative ions, induced by the dipole moment of the encapsulated molecule, these cages act as dipole-field amplifiers, different from what is observed in carbon fullerene nanocages, which exhibit instead a pronounced screening effect. Here we extend the study to another class of nanostructures: the nanotubes. Using first-principles techniques based on density functional theory, we studied the properties of endohedral nanotubes obtained by encapsulation of a water molecule or a linear HF molecule. A detailed analysis of the effective dipole moment of the complexes and of the electronic charge distribution suggests that screening effects crucially depend not only on the nature of the intramolecular bonds but also on the size and the shape of the nanotubes and on the specific encapsulated molecule. As observed in endohedral nanocages, screening is maximum in covalent-bond carbon nanotubes, while it is reduced in partially ionic nanotubes, and an antiscreening effect is observed in some ionic nanotubes. However, in this case, the scenario is more complex than in corresponding ionic nanocages. In fact the specific geometric structure of alkali-halide nanotubes turns out to be crucial for determining the screening/antiscreening behavior: while nanotubes with octagonal transversal section can exhibit an antiscreening effect, which quantitatively depends on the number of layers in the longitudinal direction, instead nanotubes with dodecagonal section are always characterized by a reduction of the total dipole moment so that a screening behavior is observed. Our results show that, even in nanotube structures, in principle one can tune the dipole moment and generate electrostatic fields at the nanoscale without the aid of external potentials.

Universal superdiffusion of random walks in media with embedded fractal networks of low diffusivity.

Diffusion in composite media with high contrasts between diffusion coefficients in fractal sets of inclusions and in their embedding matrices is modeled by lattice random walks (RWs) with probabilities p<1 of hops from fractal sites and 1 from matrix sites. Superdiffusion is predicted in time intervals that depend on p and with diffusion exponents that depend on the dimensions of matrix (E) and fractal (D_{F}) as ν=1/(2+D_{F}-E). This contrasts with the nonuniversal subdiffusion of RWs confined to fractal media. Simulations with four fractals show the anomaly at several time decades for p≲10^{-3} and the crossover to the asymptotic normal diffusion. These results show that superdiffusion can be observed in isotropic RWs with finite moments of hop length distributions and allow the estimation of the dimension of the inclusion set from the diffusion exponent. However, displacements within single trajectories have normal scaling, which shows transient ergodicity breaking.

Debiasing Welch’s method for spectral density estimation

Abstract Welch’s method provides an estimator of the power spectral density that is statistically consistent. This is achieved by averaging over periodograms calculated from overlapping segments of a time series. For a finite-length time series, while the variance of the estimator decreases as the number of segments increases, the magnitude of the estimator’s bias increases: a bias-variance trade-off ensues when setting the segment number. We address this issue by providing a novel method for debiasing Welch’s method that maintains the computational complexity and asymptotic consistency, and leads to improved finite-sample performance. Theoretical results are given for fourth-order stationary processes with finite fourth-order moments and an absolutely convergent fourth-order cumulant function. The significant bias reduction is demonstrated with numerical simulation and an application to real-world data. Our estimator also permits irregular spacing over frequency and we demonstrate how this may be employed for signal compression and further variance reduction. The code accompanying this work is available in R and python.

On the sieve M-estimation for a special bilinear time series model with time-functional variance noises

. The bilinear model has been frequently used in fields like control theory and economics to model seismic data. In this article, time-functional variance (TFV) noises are embedded into a specific bilinear model. We propose a generalized autoregressive conditional heteroskedasticity-type maximum likelihood estimator (GMLE) with a sieve method and then provide various inferential techniques based on this GMLE. It is shown that under the finite fourth moment of errors, the GMLE is consistent and asymptotically normally distributed. A simulation study and analysis of real data are additionally carried out to evaluate GMLE’s finite sample performance.

Probing the Magnetic Ground State of Ba2YIrO6: Impact of Nonmagnetic Dopants and Spin-Orbit Coupling.

Strong spin-orbit coupling (SOC) in iridates has long been predicted to lead to exotic electronic and magnetic ground states. Ba2YIrO6 (BYIO) has attracted particular attention due to the expectation of a Jeff = 0 state for Ir5+ ions under the jj-coupling scheme. However, controversies surround the actual realization of this state, as finite magnetic moments are consistently observed experimentally. We present a multi-physics study of this system by progressively introducing nonmagnetic Sb5+ ions in place of Ir5+ (Ba2YIr​1-ySb​yO​6, BYISO). Despite similar charge and ionic radii, Sb​5+ doping appears highly inhomogeneous, coexisting with a fraction of nearly pure BYIO regions, as confirmed by X-ray diffraction (XRD). This aligns with observations in related compounds. While inhomogeneity creates uncertainty, the doped majority phases offer valuable insights. It is relevant that the inclusion of even small amounts of Sb​5+ (10-20%) leads to a rise in magnetization. This strengthens our previous suggestion that magnetic Ir ions form dynamic singlets in BYIO, resulting in a near-nonmagnetic background. The observed moment enhancement with nonmagnetic doping supports the breakdown of these singlets. Furthermore, the magnetization steadily increases with an increasing Sb​5+ content, contradicting the anticipated approach towards the Jeff = 0 state with increased SOC due to reduced hopping between Ir​5+ ions. This reinforces the presence of individual Ir​5+ moments. Overall, our findings suggest that Ba​2YIrO​6 might not possess sufficiently strong SOC to be solely described within the jj-coupling picture, paving the way for further investigation.

Simple and Explicit Bounds for Multiserver Queues with 11−ρ Scaling

We consider the first-come-first-serve (FCFS) [Formula: see text] queue and prove the first simple and explicit bounds that scale as [Formula: see text] under only the assumption that interarrival times have finite second moment, and service times have finite [Formula: see text] moment for some [Formula: see text]. Here, ρ denotes the corresponding traffic intensity. Conceptually, our results can be viewed as a multiserver analogue of Kingman’s bound. Our main results are bounds for the tail of the steady-state queue length and the steady-state probability of delay. The strength of our bounds (e.g., in the form of tail decay rate) is a function of how many moments of the service distribution are assumed finite. Our bounds scale gracefully, even when the number of servers grows large and the traffic intensity converges to unity simultaneously, as in the Halfin-Whitt scaling regime. Some of our bounds scale better than [Formula: see text] in certain asymptotic regimes. In these same asymptotic regimes, we also prove bounds for the tail of the steady-state number in service. Our main proofs proceed by explicitly analyzing the bounding process that arises in the stochastic comparison bounds of Gamarnik and Goldberg for multiserver queues. Along the way, we derive several novel results for suprema of random walks and pooled renewal processes, which may be of independent interest. We also prove several additional bounds using drift arguments (which have much smaller prefactors) and point out a conjecture that would imply further related bounds and generalizations. We also show that when all moments of the service distribution are finite and satisfy a mild growth rate assumption, our bounds can be strengthened to yield explicit tail estimates decaying as [Formula: see text], with [Formula: see text], depending on the growth rate of these moments. Funding: Financial support from the National Science Foundation [Grant 1333457] is gratefully acknowledged. Supplemental Material: The supplemental appendix is available at https://doi.org/10.1287/moor.2022.0131 .

Smallest denominators

AbstractWe establish higher dimensional versions of a recent theorem by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405–1413] on the expected value of the smallest denominator of rational points in a randomly shifted interval of small length, and of the closely related 1977 Kruyswijk–Meijer conjecture recently proved by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305]. We express the distribution of smallest denominators in terms of the void statistics of multidimensional Farey fractions and prove convergence of the distribution function and certain finite moments. The latter was previously unknown even in the one‐dimensional setting. We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130–149] as well as new results on pigeonhole statistics.