Infinite Moment Research Articles

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.

Stochastic wave equation with heavy-tailed noise: Uniqueness of solutions and past light-cone property

In this article, we study the stochastic wave equation in spatial dimensions d≤2 with multiplicative Lévy noise that can have infinite pth moments. Using the past light-cone property of the wave equation, we prove the existence and uniqueness of a solution, considering only the p-integrability of the Lévy measure ν for the region corresponding to the small jumps of the noise. For d=1, there are no restrictions on ν. For d=2, we assume that there exists a value p∈(0,2) for which ∫{|z|≤1}|z|pν(dz)<+∞.

Understanding magnetic fields, for NMR/MRI

The basic physics of magnetic fields is presented for a target audience of NMR workers. This group often does not have formal training in electromagnetism, but could benefit from an understanding of a selected subset of topics. The focus here is on a relatively non-mathematical view, intended to deliver an intuitive understanding of the topic. The covered topics start with the fields arising from simple idealized currents, including long straight wires, short flat coils, infinite current sheets, long solenoids, and magnetic dipole moments. The generation of field gradients and shim fields is discussed. All of these can be unified by considering the divergence and curl of the magnetic field. Magnetic materials are treated, both linear magnetizable materials (including the sample itself) and permanently magnetized materials; the approaches of equivalent currents and Ampere's theorem for magnetic circuits are presented.

Log determinant of large correlation matrices under infinite fourth moment

Log determinant of large correlation matrices under infinite fourth moment

Correction to: “Impulsive problems on the half-line with infinite impulse moments” by Feliz Minhós, 57(1):69–79, January, 2017

Correction to: “Impulsive problems on the half-line with infinite impulse moments” by Feliz Minhós, 57(1):69–79, January, 2017

Small mass limit for stochastic interacting particle systems with Lévy noise and linear alignment force.

We study the small mass limit in mean field theory for an interacting particle system with non-Gaussian Lévy noise. When the Lévy noise has a finite second moment, we obtain the limit equation with convergence rate ε+1/εN, by taking first the mean field limit N→∞ and then the small mass limit ε→0. If the order of the two limits is exchanged, the limit equation remains the same but has a different convergence rate ε+1/N. However, when the Lévy noise is α-stable, which has an infinite second moment, we can only obtain the limit equation by taking first the small mass limit and then the mean field limit, with the convergence rate 1/Nα-1+1/Np2+εp/α where p∈(1,α). This provides an effectively limit model for an interacting particle system under a non-Gaussian Lévy fluctuation, with rigorous error estimates.

A hypothesis test for the domain of attraction of a random variable

In this work, we address the problem of detecting whether a sampled probability distribution of a random variable V has infinite first moment. This issue is notably important when the sample results from complex numerical simulation methods. For example, such a situation occurs when one simulates stochastic particle systems with complex and singular McKean–Vlasov interaction kernels. As stated, the detection problem is ill-posed. We thus propose and analyze an asymptotic hypothesis test for independent copies of a given random variable which is supposed to belong to an unknown domain of attraction of a stable law. The null hypothesis H0 is: ‘√V is in the domain of attraction of the Normal law’ and the alternative hypothesis is H1: ‘X is in the domain of attraction of a stable law with index smaller than 2’. Our key observation is that X cannot have a finite second moment when H0 is rejected (and therefore H1 is accepted). Surprisingly, we find it useful to derive our test from the statistics of random processes. More precisely, our hypothesis test is based on a statistic which is inspired by methodologies to determine whether a semimartingale has jumps from the observation of one single path at discrete times. We justify our test by proving asymptotic properties of discrete time functionals of Brownian bridges.

Extending the Applicability of the Pollaczek-Khinchin Formula to the Case of Infinite Service Moments

Extending the Applicability of the Pollaczek-Khinchin Formula to the Case of Infinite Service Moments

Coupled continuous time random walk with Lévy distribution jump length signifies anomalous diffusion?

The continuous time random walk (CTRW) models are highly valued for studying anomalous diffusion, a ubiquitous phenomenon in complex media whose mean squared displacement (MSD) exhibits power-law behavior. It is widely accepted that the Lévy distribution jump length of random particles with infinite second moment could be responsible for the superdiffusion phenomenon. In this paper, we study the coupled CTRW with exponentially truncated Lévy distribution jump length, deduce the corresponding integrodifferential diffusion equation, and give the propagator expression. It is interesting that we find the MSD for random particles is linearly related to time in the symmetric Lévy distribution case, and it reduces to a simple expression. Additionally, for a non-zero truncated exponent, the MSD also has a linear relationship with time, which reflects the special dependence between waiting time and jump length in this coupled CTRW model.

On a certain martingale representation and the related infinite dimensional moment problem

On a certain martingale representation and the related infinite dimensional moment problem

Distributed Recursive Estimation under Heavy-Tail Communication Noise

We consider distributed recursive estimation of an unknown vector parameter in the presence of impulsive communication noise. That is, we assume that interagent communication is subject to an additive communication noise that may have heavy-tails or is contaminated with outliers. To combat this effect, within the class of consensus+innovations distributed estimators, we introduce for the first time a nonlinearity in the consensus update. We allow for a general class of nonlinearities that subsumes, e.g., the sign function or componentwise saturation function. For the general nonlinear estimator and a general class of additive communication noises—that may have infinite moments of order higher than one—we establish almost sure convergence to the parameter . We further prove asymptotic normality and evaluate the corresponding asymptotic covariance. These results reveal interesting tradeoffs between the negative effect of “loss of information” due to incorporation of the nonlinearity and the positive effect of communication noise reduction. We also demonstrate and quantify benefits of introducing the nonlinearity in high-noise (low signal-to-noise ratio) and heavy-tail communication noise regimes.

Rank-based testing for semiparametric VAR models: A measure transportation approach

We develop a class of tests for semiparametric vector autoregressive (VAR) models with unspecified innovation densities, based on the recent measure-transportation-based concepts of multivariate {\it center-outward ranks} and {\it signs}. We show that these concepts, combined with Le Cam's asymptotic theory of statistical experiments, yield novel testing procedures, which (a)~are valid under a broad class of innovation densities (possibly non-elliptical, skewed, and/or with infinite moments), (b)~are optimal (locally asymptotically maximin or most stringent) at selected ones, and (c) are robust against additive outliers. In order to do so, we establish a H\' ajek asymptotic representation result, of independent interest, for a general class of center-outward rank-based serial statistics. As an illustration, we consider the problems of testing the absence of serial correlation in multiple-output and possibly non-linear regression (an extension of the classical Durbin-Watson problem) and the sequential identification of the order $p$ of a vector autoregressive (VAR($p$)) model. A Monte Carlo comparative study of our tests and their routinely-applied Gaussian competitors demonstrates the benefits (in terms of size, power, and robustness) of our methodology; these benefits are particularly significant in the presence of asymmetric and leptokurtic innovation densities. A real data application concludes the paper.

Correlation versus co-fractality: Evidence from foreign-exchange-rate variances

The concept of correlation appears to be the cornerstone of modern finance as it is applied in almost all finance-related research studies. However, Fama (1963) argued that “if the [population] variance is infinite, other statistical tools (e.g., least-squares regression) which are based on the assumption of finite variance will, at best, be considerably weakened and may in fact give very misleading answers” (p. 421). This study shows variances of foreign exchange rates to be governed by power laws with a tail exponent of α < 3, suggesting infinite second moments. We derive a new concept to measure dependencies between power-law processes with this tail exponent, which we term co-fractality. We show that risk diversification based on the concept of correlation indeed gives misleading results. Notably, foreign-exchange-rate variances lacking co-fractality in our earlier subsample do not show evidence for co-fractality in our later subsample. We argue that co-fractality, as opposed to correlation, should be used to measure the dependency between processes governed by power laws.

Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of random variables with infinite moment and its applications

In this work, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of widely orthant dependent random variables is established under mild conditions including infinite variance. As applications, some new results such as the Cesàro strong law of large numbers, the strong consistency of the least squares estimator in multiple linear regression models, and the strong consistency of the wavelet estimator in nonparametric regression models are presented. Simulation studies are also provided to support the theoretical results.

A Family of Discrete Distributions with Infinite Moments

A Family of Discrete Distributions with Infinite Moments

On the consistency of incomplete U-statistics under infinite second-order moments

We derive a consistency result, in the L1-sense, for incomplete U-statistics in the non-standard case where the kernel at hand has infinite second-order moments. Assuming that the kernel has finite moments of order p(≥1), we obtain a bound on the L1 distance between the incomplete U-statistic and its Dirac weak limit, which allows us to obtain, for any fixed p, an upper bound on the consistency rate. Our results hold for most classical sampling schemes that are used to obtain incomplete U-statistics.

Necessary and sufficient conditions for convergence to the semicircle distribution

We consider random Hermitian matrices with independent upper triangular entries. Wigner’s semicircle law says that under certain additional assumptions, the empirical spectral distribution converges to the semicircle distribution. We characterize convergence to semicircle in terms of the variances of the entries, under natural assumptions such as the Lindeberg condition. The result extends to certain matrices with entries having infinite second moments. As a corollary, another characterization of semicircle convergence is given in terms of convergence in distribution of the row sums to the standard normal distribution.

Limit theorems for linear random fields with innovations in the domain of attraction of a stable law

In this paper we study the convergence in distribution and the local limit theorem for the partial sums of linear random fields with i.i.d. innovations that have infinite second moment and belong to the domain of attraction of a stable law with index 0<α≤2 under the condition that the innovations are centered if 1<α≤2 and are symmetric if α=1. We establish these two types of limit theorems as long as the linear random fields are well-defined, the coefficients are either absolutely summable or not absolutely summable.

A NOTE ON OPEN QUESTIONS ASKED TO ANALYSIS AND NUMERICS CONCERNING THE HAUSDORFF MOMENT PROBLEM

Abstract We address facts and open questions concerning the degree of ill-posedness of the composite Hausdorff moment problem aimed at the recovery of a function x ∈ L 2 (0, 1) from elements of the infinite dimensional sequence space ` 2 that characterize moments applied to the antiderivative of x. This degree, unknown by now, results from the decay rate of the singular values of the associated compact forward operator A, which is the composition of the compact simple integration operator mapping in L 2 (0, 1) and the non-compact Hausdorff moment operator B(H) mapping from L 2 (0, 1) to ` 2 . There is a seeming contradiction be- tween (a) numerical computations, which show (even for large n) an exponential decay of the singular values for n-dimensional matrices obtained by discretizing the operator A, and (b) a strongly limited smoothness of the well-known kernel k of the Hilbert-Schmidt operator A∗A. Fact (a) suggests severe ill-posedness of the infinite dimensional Hausdorff moment problem, whereas fact (b) lets us expect the opposite, because exponential ill-posedness oc- curs in common just for C∞-kernels k. We recall arguments for the possible occurrence of a polynomial decay of the singular values of A, even if the numerics seems to be against it, and discuss some issues in the numerical approximation of non-compact operators.

Stable Central Limit Theorems for Super Ornstein-Uhlenbeck Processes, II

This paper is a continuation of our recent paper (Electron. J. Probab., 24(141), (2019)) and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes (Xt)t≥0 with branching mechanisms of infinite second moments. In the aforementioned paper, we proved stable central limit theorems for Xt(f) for some functions f of polynomial growth in three different regimes. However, we were not able to prove central limit theorems for Xt(f) for all functions f of polynomial growth. In this note, we show that the limiting stable random variables in the three different regimes are independent, and as a consequence, we get stable central limit theorems for Xt(f) for all functions f of polynomial growth.