Infinite Moment Research Articles

Cluster Tails for Critical Power-Law Inhomogeneous Random Graphs

Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained in Bhamidi et al. (Ann Probab 40:2299–2361, 2012). It was proved that when the degrees obey a power law with exponent tau in (3,4), the sequence of clusters ordered in decreasing size and multiplied through by n^{-(tau -2)/(tau -1)} converges as nrightarrow infty to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel (J Combin Theory Ser B 82(2):237–269, 2001) for the Erdős–Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

Information Measures, Inequalities and Performance Bounds for Parameter Estimation in Impulsive Noise Environments

Recent studies found that many channels are affected by additive noise that is impulsive in nature and is best explained by heavy-tailed symmetric alpha-stable distributions. Dealing with impulsive noise environments comes with an added complexity with respect to the standard Gaussian environment: the alpha-stable probability density functions do not possess closed-form expressions except in few special cases. Furthermore, they have an infinite second moment and the “nice” Hilbert space structure of the space of random variables having a finite second moment is lost along with its tools and methodologies. This is indeed the case in estimation theory, where classical tools to quantify the performance of an estimator are tightly related to the assumption of having finite variance variables. In alpha-stable environments, expressions, such as the mean square error and the Cramer–Rao bound, are hence problematic. In this paper, we tackle the parameter-estimation problem in the impulsive noise environments and develop novel tools that are tailored to the alpha-stable and heavy-tailed noise environments, tools that coincide with the standard ones adopted in the Gaussian setup, namely, a generalized “power” measure and a generalized Fisher information. We generalize known information inequalities commonly used in the Gaussian context: the de Bruijn identity, the Fisher information inequality, the isoperimetric inequality for entropies and the Cramer–Rao bound. Additionally, we derive upper bounds on the differential entropy of independent sums having a stable component. Intermediately, the new power measure is used to shed some light on the additive alpha-stable noise channel capacity in a setup that generalizes the linear average power constrained additive white Gaussian noise channel. Our theoretical findings are paralleled with numerical evaluations of various quantities and bounds using developed MATLAB packages.

Skewness and kurtosis analysis for non-Gaussian distributions

In this paper we address a number of pitfalls regarding the use of kurtosis as a measure of deviations from the Gaussian. We treat kurtosis in both its standard definition and that which arises in q-statistics, namely q-kurtosis. We have recently shown that the relation proposed by Cristelli et al. (2012) between skewness and kurtosis can only be verified for relatively small data sets, independently of the type of statistics chosen; however it fails for sufficiently large data sets, if the fourth moment of the distribution is finite. For infinite fourth moments, kurtosis is not defined as the size of the data set tends to infinity. For distributions with finite fourth moments, the size, N, of the data set for which the standard kurtosis saturates to a fixed value, depends on the deviation of the original distribution from the Gaussian. Nevertheless, using kurtosis as a criterion for deciding which distribution deviates further from the Gaussian can be misleading for small data sets, even for finite fourth moment distributions. Going over to q-statistics, we find that although the value of q-kurtosis is finite in the range of 0<q<3, this quantity is not useful for comparing different non-Gaussian distributed data sets, unless the appropriate q value, which truly characterizes the data set of interest, is chosen. Finally, we propose a method to determine the correct q value and thereby to compute the q-kurtosis of q-Gaussian distributed data sets.

Shear stresses in rectangular panels of ship structures in the calculations according to Reissner theory

This study calculates and analyzes torsion moments of a rectangular panel with clamped edges as an element of ship structures under the action of uniform pressure with allowance for transverse shear deformation and examines the contribution of the corresponding shear stresses to the general stress state. In order to solve this problem, the method of infinite superposition of corrective functions of bending and stresses is applied. It involves an iterative process of mutually correcting the discrepancies from the said functions while meeting all boundary conditions. A particular solution for the bending function in the form of a quadratic polynomial is chosen as the initial approximation. It is established that torsion moment series diverge at the corner points of the plate going into infinity, which yields infinite values for the shear stresses at these points as well. Results of torsion moment calculation for square plates with different width ratios are provided. A 3D distribution diagram of moments is obtained. The computational experiment confirms the correctness of theoretical conclusions about infinite torsion moments at the corner points of the plate. Comparison with bending moments shows that torsion moments cannot be ignored during the assessment of the stress-strain state. The behavior of torsion moments near the corner points is qualitatively different from the simplified Kirchhoff theory, where they turn into zero.

Fractal scaling analysis of groundwater dynamics in confined aquifers

Abstract. Groundwater closely interacts with surface water and even climate systems in most hydroclimatic settings. Fractal scaling analysis of groundwater dynamics is of significance in modeling hydrological processes by considering potential temporal long-range dependence and scaling crossovers in the groundwater level fluctuations. In this study, it is demonstrated that the groundwater level fluctuations in confined aquifer wells with long observations exhibit site-specific fractal scaling behavior. Detrended fluctuation analysis (DFA) was utilized to quantify the monofractality, and multifractal detrended fluctuation analysis (MF-DFA) and multiscale multifractal analysis (MMA) were employed to examine the multifractal behavior. The DFA results indicated that fractals exist in groundwater level time series, and it was shown that the estimated Hurst exponent is closely dependent on the length and specific time interval of the time series. The MF-DFA and MMA analyses showed that different levels of multifractality exist, which may be partially due to a broad probability density distribution with infinite moments. Furthermore, it is demonstrated that the underlying distribution of groundwater level fluctuations exhibits either non-Gaussian characteristics, which may be fitted by the Lévy stable distribution, or Gaussian characteristics depending on the site characteristics. However, fractional Brownian motion (fBm), which has been identified as an appropriate model to characterize groundwater level fluctuation, is Gaussian with finite moments. Therefore, fBm may be inadequate for the description of physical processes with infinite moments, such as the groundwater level fluctuations in this study. It is concluded that there is a need for generalized governing equations of groundwater flow processes that can model both the long-memory behavior and the Brownian finite-memory behavior.

Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from n independent observations of a p-dimensional time series with iid components converge almost surely to (1+γ)2 and (1−γ)2, respectively, as n→∞, if p∕n→γ∈(0,1] and the truncated variance of the entry distribution is “almost slowly varying”, a condition we describe via moment properties of self-normalized sums. Moreover, the empirical spectral distributions of these sample correlation matrices converge weakly, with probability 1, to the Marčenko–Pastur law, which extends a result in Bai and Zhou (2008). We compare the behavior of the eigenvalues of the sample covariance and sample correlation matrices and argue that the latter seems more robust, in particular in the case of infinite fourth moment. We briefly address some practical issues for the estimation of extreme eigenvalues in a simulation study.In our proofs we use the method of moments combined with a Path-Shortening Algorithm, which efficiently uses the structure of sample correlation matrices, to calculate precise bounds for matrix norms. We believe that this new approach could be of further use in random matrix theory.

Lévy-Khintchine random matrices and the Poisson weighted infinite skeleton tree

We study a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as adjacency matrices of Erdős-Rényi random graphs with p n ∼ 1 n p_n\sim \frac 1 n . Our n × n n\times n random matrices have real entries which are i.i.d. up to symmetry. The distribution of entries depends on n n , and we require row sums to converge in distribution. It is then well-known that the limit distribution must be infinitely divisible. We show that a limiting empirical spectral distribution (LSD) exists and, via local weak convergence of associated graphs, that the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges called “cords to infinity”. One example covered by the results are matrices with i.i.d. entries having infinite second moments but normalized to be in the Gaussian domain of attraction. In this case, the limiting graph is N \mathbb {N} rooted at 1 1 , so the LSD is the semicircle law.

Empirical likelihood for compound Poisson processes under infinite second moment

ABSTRACTThe compound Poisson process has been widely used in many fields, for example, physics, engineering, finance, and so on. Regarding the process, the average number, namely t− 1E[SN(t)] = λμ, attracts lots of interests. In this article, we derive the limiting behavior of the log empirical likelihood ratio statistic for λμ when the population is in the domain of attraction of normal law. The simulation studies confirm the theoretical result.

Optimization problem for a portfolio with an illiquid asset: Lie group analysis

Management of a portfolio that includes an illiquid asset is an important problem of modern mathematical finance. One of the ways to model illiquidity among others is to build an optimization problem and assume that one of the assets in a portfolio cannot be sold until a certain finite, infinite or random moment of time. This approach arises a certain amount of models that are actively studied at the moment.Working in the Merton's optimal consumption framework with continuous time we consider an optimization problem for a portfolio with an illiquid, a risky and a risk-free asset. Our goal in this paper is to carry out a complete Lie group analysis of PDEs describing value function and investment and consumption strategies for a portfolio with an illiquid asset that is sold in an exogenous random moment of time with a prescribed liquidation time distribution. The problem of such type leads to three dimensional nonlinear Hamilton–Jacobi–Bellman (HJB) equations. Such equations are not only tedious for analytical methods but are also quite challenging form a numeric point of view. To reduce the three-dimensional problem to a two-dimensional one or even to an ODE one usually uses some substitutions, yet the methods used to find such substitutions are rarely discussed by the authors.We find the admitted Lie algebra for a broad class of liquidation time distributions in cases of HARA and log utility functions and formulate corresponding theorems for all these cases. We use found Lie algebras to obtain reductions of the studied equations. Several of similar substitutions were used in other papers before whereas others are new to our knowledge. This method gives us the possibility to provide a complete set of non-equivalent substitutions and reduced equations.

Empirical likelihood ratio under infinite second moment

ABSTRACTIn this article, we show that the log empirical likelihood ratio statistic for the population mean converges in distribution to χ2(1) as n → ∞ when the population is in the domain of attraction of normal law but has infinite variance. The simulation results show that the empirical likelihood ratio method is applicable under the infinite second moment condition.

Heavy-Tailed Analogues of the Covariance Matrix for ICA

Independent Component Analysis (ICA) is the problem of learning a square matrix A, given samples of X = AS, where S is a random vector with independent coordinates. Most existing algorithms are provably efficient only when each Si has finite and moderately valued fourth moment. However, there are practical applications where this assumption need not be true, such as speech and finance. Algorithms have been proposed for heavy-tailed ICA, but they are not practical, using random walks and the full power of the ellipsoid algorithm multiple times. The main contributions of this paper are (1) A practical algorithm for heavy-tailed ICA that we call HTICA. We provide theoretical guarantees and show that it outperforms other algorithms in some heavy-tailed regimes, both on real and synthetic data. Like the current state-of-the-art, the new algorithm is based on the centroid body (a first moment analogue of the covariance matrix). Unlike the state-of-the-art, our algorithm is practically efficient. To achieve this, we use explicit analytic representations of the centroid body, which bypasses the use of the ellipsoid method and random walks. (2) We study how heavy tails affect different ICA algorithms, including HTICA. Somewhat surprisingly, we show that some algorithms that use the covariance matrix or higher moments can successfully solve a range of ICA instances with infinite second moment. We study this theoretically and experimentally, with both synthetic and real-world heavy-tailed data.

UNIFORM CONVERGENCE RATES OVER MAXIMAL DOMAINS IN STRUCTURAL NONPARAMETRIC COINTEGRATING REGRESSION

This paper presents uniform convergence rates for kernel regression estimators, in the setting of a structural nonlinear cointegrating regression model. We generalise the existing literature in three ways. First, the domain to which these rates apply is much wider than the domains that have been considered in the existing literature, and can be chosen so as to contain as large a fraction of the sample as desired in the limit. Second, our results allow the regression disturbance to be serially correlated, and cross-correlated with the regressor; previous work on this problem (of obtaining uniform rates) having been confined entirely to the setting of an exogenous regressor. Third, we permit the bandwidth to be data-dependent, requiring it to satisfy only certain weak asymptotic shrinkage conditions. Our assumptions on the regressor process are consistent with a very broad range of departures from the standard unit root autoregressive model, allowing the regressor to be fractionally integrated, and to have an infinite variance (and even infinite lower-order moments).

Distribution of spectral linear statistics on random matrices beyond the large deviation function—Wigner time delay in multichannel disordered wires

An invariant ensemble of N × N random matrices can be characterised by a joint distribution for eigenvalues . The distribution of linear statistics, i.e. of quantities of the form where f(x) is a given function, appears in many physical problems. In the limit, L scales as , where the scaling exponent η depends on the ensemble and the function f(x). Its distribution can be written in the form , where is the Dyson index. The Coulomb gas technique naturally provides the large deviation function , which can be efficiently obtained thanks to a ‘thermodynamic identity’ introduced earlier. We conjecture the pre-exponential function . We check our conjecture on several well controlled cases within the Laguerre and the Jacobi ensembles. Then we apply our main result to a situation where the large deviation function has no minimum (and L has infinite moments): this arises in the statistical analysis of the Wigner time delay for semi-infinite multichannel disordered wires (Laguerre ensemble). The statistical analysis of the Wigner time delay then crucially depends on the pre-exponential function , which ensures the decay of the distribution for large argument.

Spectral covariance and limit theorems for random fields with infinite variance

In the paper, we continue to investigate measures of dependence for random variables with infinite variance. For random variables with regularly varying tails, we introduce a general class of such measures, which includes the codifference and the spectral covariance. In particular, we investigate the α -spectral covariance, a new measure from this general class, for linear random fields with infinite second moment. Under some conditions on the filter of a linear random field, we investigate asymptotic properties of the α -spectral covariance for linear random fields with infinite variance. We also provide an application of spectral covariances for limit theorems for stationary and associated random fields with infinite variance.

Law of the iterated logarithm type results for random vectors with infinite second moments

This survey paper is an extended version of the author’s presentation at the conference in honor of Professor F. Thomas Bruss at the occasion of his retirement as Chair of Math´ematiques G´en´erales from the Unversit´e Libre de Bruxelles which was held September 9-11, 2015 in Brussels. I first present some results generalizing the classical Hartman-Wintner law of the iterated logarithm to 1-dimensional variables with infinite second moments and then I show how these results can be further extended to the d-dimensional setting. Finally, I look at general functional law of the iterated logarithm type results.

A CLT for martingale transforms with infinite variance

We provide a CLT for martingale transforms that holds even when the second moments are infinite. Compared to an analogous result in Hall and Yao (2003) we impose minimal assumptions and utilize the Principle of Conditioning to verify a modified version of Lindeberg’s condition. When the variance is infinite, the rate of convergence, which we allow to be matrix valued, is slower than n and depends on the rate of divergence of the truncated second moments. In many cases it can be consistently estimated. A major application concerns the characterization of the rate and the limiting distribution of the Gaussian QMLE in the case of GARCH type models with infinite fourth moments for the innovation process. The results are particularly useful in the case of the EGARCH(1,1) model as we show that the usual limit theory is still valid without any further parameter restrictions when we relax the assumption for finite fourth moments of the innovation process.

On Barnes Beta Distributions and Applications to the Maximum Distribution of the 2D Gaussian Free Field

A new family of Barnes beta distributions on $(0, \infty)$ is introduced and its infinite divisibility, moment determinacy, scaling, and factorization properties are established. The Morris integral probability distribution is constructed from Barnes beta distributions of types $(1,0)$ and $(2,2),$ and its moment determinacy and involution invariance properties are established. For application, the maximum distributions of the 2D gaussian free field on the unit interval and circle with a non-random logarithmic potential are conjecturally related to the critical Selberg and Morris integral probability distributions, respectively, and expressed in terms of sums of Barnes beta distributions of types $(1,0)$ and $(2,2).$

Invariance principles and almost sure central limit theorem for the error variance estimator in linear models

ABSTRACTIn this article, we establish the invariance principles for the error variance estimator in linear models, which weaken the conditions for the errors in Chen (1980) from finite fourth moment to infinite fourth moment. Furthermore, the almost sure central limit theorem of the error variance estimator is also obtained under the assumption for the errors of infinite fourth moment.

Estimation and tests for TGTACH$\bm{(1, 1)}$ models with heavy-tailed errors: A uniform framework

This paper studies the estimation and tests of heavy-tailed TGARCH$(1, 1)$ models based on quasi-maximum likelihood estimation (QMLE) and percentile-$t$ subsample bootstrap method, where heavy-tail means that the distribution of the square errors are in the domain of attraction of a stable law with infinite fourth moment. Under regular conditions, we first establish that the QMLE of ARCH and GARCH coefficients are consistent and their asymptotic distribution is non-Gaussian but is some stable law with exponent $\kappa\in (1, 2)$ despite of whether the heavy tailed TGARCH model is strictly stationary or not. However, the QMLE of the location parameter is consistent only when the model is strictly stationary. Then we propose tests for strict stationarity and symmetry for heavy-tailed TGARCH$(1, 1)$ models based the above asymptotic results and percentile-$t$ subsample bootstrap method to overcome the difficulty that the scale parameter and asymptotic distribution depend on the unknown exponent. These tests can be used in the universal parameter space. Finally, we conduct a simulation study through Monte Carlo methods to illustrate the performance of the proposed methods with finite sample sizes and then give an empirical analysis for 5-year China Treasury Bond Futures to illustrate the application of our methods.

A New Tight Upper Bound on the Entropy of Sums

We consider the independent sum of a given random variable with a Gaussian variable and an infinitely divisible one. We find a novel tight upper bound on the entropy of the sum which still holds when the variable possibly has an infinite second moment. The proven bound has several implications on both information theoretic problems and infinitely divisible noise channels’ transmission rates.