Let $X_N$ be an $N\ts N$ random symmetric matrix with independent equidistributed entries. If the law $P$ of the entries has a finite second moment, it was shown by Wigner \cite{wigner} that the empirical distribution of the eigenvalues of $X_N$, once renormalized by $\sqrt{N}$, converges almost surely and in expectation to the so-called semicircular distribution as $N$ goes to infinity. In this paper we study the same question when $P$ is in the domain of attraction of an $α$-stable law. We prove that if we renormalize the eigenvalues by a constant $a_N$ of order $N^{\frac{1}α}$, the corresponding spectral distribution converges in expectation towards a law $μ_α$ which only depends on $α$. We characterize $μ_α$ and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero.
We study the free energy of a mean-field spin glass whose coupling distribution has power law tails. Under the assumption that the couplings have infinite variance and finite mean, we show that the thermodynamic limit of the quenched free energy exists, and that the free energy is self-averaging.
We prove that the eigenvectors associated to small enough eigenvalues of an heavy-tailed symmetric random matrix are delocalized with probability tending to one as the size of the matrix grows to infinity. The delocalization is measured thanks to a simple criterion related to the inverse participation ratio which computes an average ratio of L4 and L2-norms of vectors. In contrast, as a consequence of a previous result, for random matrices with sufficiently heavy tails, the eigenvectors associated to large enough eigenvalues are localized according to the same criterion. The proof is based on …
We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure $μ$ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N by N symmetric matrix $Y_N^σ$ whose (i,j) entry is $σ(i/N,j/N)X_{ij}$ where $(X_{ij}, 0<i<j+1<\infty)$ is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an $α$-stable law, $0<α<2$, and $σ$ is a deterministic function. For a random diagonal $D_N$ independent of $Y_N^σ$ and with appropriate rescaling $a_N$, we prove that the distribution $μ$ of $a_…
We consider a Wigner matrix $A$ with entries tail decaying as $x^{-α}$ with $2<α<4$ for large $x$ and study fluctuations of linear statistics $N^{-1}\operatorname{Tr}φ(A)$. The behavior of such fluctuations has been understood for both heavy-tailed matrices (i.e. $α< 2$) and light-tailed matrices (i.e. $α> 4$). This paper fills in the gap of understanding for $2<α<4$. We find that while linear spectral statistics for heavy-tailed matrices have fluctuations of order $N^{-1/2}$ and those for light-tailed matrices have fluctuations of order $N^{-1}$, the linear spectral statistics for half-heavy-…
In this paper the question of the extent to which truncated heavy tailed random vectors, taking values in a Banach space, retain the characteristic features of heavy tailed random vectors, is answered from the point of view of the central limit theorem.
This paper investigates an upper bound of the operator norm for sub-Gaussian tailed random matrices. A lot of attention has been put on uniformly bounded sub-Gaussian tailed random matrices with independent coefficients. However, little has been done for sub-Gaussian tailed random matrices whose matrix coefficients variance are not equal or for matrix for which coefficients are not independent. This is precisely the subject of this paper. After proving that random matrices with uniform sub-Gaussian tailed independent coefficients satisfy the Tracy Widom bound, that is, their matrix operator no…
We present an overview of possible reasons for the appearance of heavy-tailed distributions in applications to the natural sciences. These distributions include the laws of Pareto, Lotka, and some new ones. The reasons are illustrated using suitable toy models. Keywords: heavy-tailed distributions; Pareto law; Lotka law; Zipf law; probability generating function.
We study tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.
We analyze the largest eigenvalue statistics of m-dependent heavy-tailed Wigner matrices as well as the associated sample covariance matrices having entry-wise regularly varying tail distributions with parameter $0<α<4$. Our analysis extends results in the previous literature for the corresponding random matrices with independent entries above the diagonal, by allowing for m-dependence between the entries of a given matrix. We prove that the limiting point process of extreme eigenvalues is a Poisson cluster process.
We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in \cite{Sos1}, we prove that, in the absence of the fourth moment, the top eigenvalues behave, in the limit, as the largest entries of the matrix.
In this work we characterize the local asymptotic self-similarity of harmonizable fractional Lévy motions in the heavy tailed case. The corresponding tangent process is shown to be the harmonizable fractional stable motion. In addition, we provide sufficient conditions for existence of harmonizable fractional Lévy motions.
The purpose of this paper is to show that the use of heavy-tailed distributions in Financial problems is theoretically baseless and can lead to significant misunderstandings. The reason for this the authors see in an incorrect interpretation of the concept of the distributional tail. In accordance with this, in applications it is necessary to use instead of the distributional tail the "smearing" of its central part. keywords: heavy tailed distributions, exponential tails, pre-limit theorems, Financial indexes
In this paper we derive exact large-buffer asymptotics for a two-class Generalized Processor Sharing (GPS) model, under the assumption that the input traffic streams generated by both classes correspond to heavy-tailed Lévy processes. Four scenarios need to be distinguished, which differ in terms of (i)~the level of heavy-tailedness of the driving Lévy processes as well as (ii)~the values of the corresponding mean rates relative to the GPS weights. The derived results are illustrated by two important special cases, in which the queues' inputs are modeled by heavy-tailed compound Poisson proces…
This article studies asymptotic approximations of ruin probabilities of multivariate random walks with heavy-tailed increments. Under our assumptions, the distributions of the increments are closely connected to multivariate subexponentiality and admit dependence between components. Keywords: subexponential distribution, ruin probability, multivariate random walk
We study the first exit times form a reduced domain of attraction of a stable fixed of the Chafee-Infante equation when perturbed by a heavy tailed Lévy noise with small intensity.
We investigate maxima of linear processes with i.i.d. heavy-tailed innovations and random coefficients. Using the point process approach we derive functional convergence of the partial maxima stochastic process in the space of non-decreasing càdlàg functions on $[0,1]$ with the Skorohod $M_{1}$ topology.
It is well-known that large deviations of random walks driven by independent and identically distributed heavy-tailed random variables are governed by the so-called principle of one large jump. We note that further subtleties hold for such random walks in the large deviation scale which we call hidden large deviation. We apply this idea in the context of queueing processes with heavy-tailed service times and study approximations of severe congestion times for (buffered) queues. We conclude with simulated examples to verify our results.
We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might …
This paper deals with symmetric random matrices whose upper diagonal entries are obtained from a linear random field with heavy tailed noise. It is shown that the maximum eigenvalue and the spectral radius of such a random matrix with dependent entries converge to the Frechét distribution after appropriate scaling. This extends a seminal result of Soshnikov(2004) when the tail index is strictly less than one.
This work deals with the divisible sandpile model when an initial configuration sampled from a heavy-tailed distribution. Extending results of Levine et al. (2015) and Cipriani et al. (2016) we determine sufficient conditions for stabilization and non-stabilization on infinite graphs. We determine furthermore that the scaling limit of the odometer on the torus is an $α$-stable random distribution.
We define heavy-tailed fractional reciprocal gamma and Fisher-Snedecor diffusions by a non-Markovian time change in the corresponding Pearson diffusions. Pearson diffusions are governed by the backward Kolmogorov equations with space-varying polynomial coefficients and are widely used in applications. The corresponding fractional reciprocal gamma and Fisher-Snedecor diffusions are governed by the fractional backward Kolmogorov equations and have heavy-tailed marginal distributions in the steady state. We derive the explicit expressions for the transition densities of the fractional reciprocal …
Several convenient methods for calculation of fractional absolute moments are given with application to heavy tailed distributions. We use techniques of fractional differentiation to obtain formulae for $E[|X-μ|^γ]$ with $1<γ<2$ and $μ\in\mathbb{R}$. The main focus is on heavy tailed distributions, several examples are given with analytical expressions of fractional absolute moments. As applications, we calculate the fractional moment errors for both prediction and parameter estimation problems.
Sub-Gaussian estimates for the natural random walk is typical of many regular fractal graphs. Subordination shows that there exist heavy tailed jump processes whose jump indices are greater than or equal to two. However, the existing machinery used to prove heat kernel bounds for such heavy tailed random walks fail in this case. In this work we extend Davies' perturbation method to obtain transition probability bounds for these anomalous heavy tailed random walks. We prove global upper and lower bounds on the transition probability density that are sharp up to constants. An important feature o…
We prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes with a certain type of negative dependence. Previously only positive dependence could be treated. The negative dependence involves cancellations of the Gaussian second order. This leads to new types of limiting processes involving stable random measures, due to heavy tails, Mittag-Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations.
Constructions of numerous approximate sampling algorithms are based on the well-known fact that certain Gibbs measures are stationary distributions of ergodic stochastic differential equations (SDEs) driven by the Brownian motion. However, for some heavy-tailed distributions it can be shown that the associated SDE is not exponentially ergodic and that related sampling algorithms may perform poorly. A natural idea that has recently been explored in the machine learning literature in this context is to make use of stochastic processes with heavy tails instead of the Brownian motion. In this pape…
We show central limit theorems (CLT) for the Stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of $α$-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike to the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent r…
For a class of stationary regularly varying and weakly dependent time series, we prove the so-called complete convergence result for the corresponding space-time point processes. As an application of our main theorem, we give a simple proof of the invariance principle for the corresponding partial maximum process.
We prove large deviation results for sums of heavy-tailed random elements in rather general convex cones being semigroups equipped with a rescaling operation by positive real numbers. In difference to previous results for the cone of convex sets, our technique does not use the embedding of cones in linear spaces. Examples include the cone of convex sets with the Minkowski addition, positive half-line with maximum operation and the family of square integrable functions with arithmetic addition and argument rescaling.
A large deviations principle is established for the joint law of the empirical measure and the flow measure of a renewal Markov process on a finite graph. We do not assume any bound on the arrival times, allowing heavy tailed distributions. In particular, the rate functional is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behavior highly different from what one may guess with a heuristic Donsker-Varadhan analysis of the problem.
In this paper, we will give a sufficient condition for a non-negative random variable $X$ to be heavy tailed by investigating the Laplace-Stieltjes transform of the probability distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. Our theorem is a kind of Tauberian theorems.
We consider asymptotics for the maximum of a modulated random walk whose increments $ξ_n^{X_n}$ are heavy-tailed. Of particular interest is the case where the modulating process $X$ is regenerative. Here we study also the maximum of the recursion given by $W_0=0$ and, for $n\ge1$, $W_n=\max(0,W_{n-1}+ξ_n^{X_n})$
The choice of the increment distribution is crucial for the random-walk Metropolis-Hastings (RWM) algorithm. In this paper we study the optimal choice in high-dimension setting among all possible increment distributions. The conclusion is rather counter intuitive, but the optimal rate of convergence is attained by the usual choice, the normal distribution as the increment distribution. In particular, no heavy-tailed increment distribution can improve the rate.
In this paper we first provide several conditional limit theorems for Lévy processes with negative drift and regularly varying tail. Then we apply them to study the asymptotic behavior of expectations of some exponential functionals of heavy-tailed Lévy processes. As the key point, we observe that the asymptotics mainly depends on the sample paths with early arrival large jump. Both the polynomial decay rate and the exact expression of the limit coefficients are given. As an application, we give an exact description for the extinction speed of continuous-state branching processes in heavy-tail…
This paper investigates the decay rate of the probability that the row sum of a triangular array of truncated heavy tailed random variables is larger than an integer (k) times the truncating threshold, as both - the number of summands and the threshold go to infinity. The method of attack for this problem is significantly different from the one where k is not an integer, and requires much sharper estimates.
This paper studies the effect of truncation on the large deviations behavior of the partial sum of a triangular array coming from a truncated power law model. Each row of the triangular array consists of i.i.d. random vectors, whose distribution matches a power law on a ball of radius going to infinity, and outside that it has a light-tailed modification. The random vectors are assumed to be R^d-valued. It turns out that there are two regimes depending on the growth rate of the truncating threshold, so that in one regime, much of the heavy tailedness is retained, while in the other regime, the…
We say that a random variable is $light$-$tailed$ if moments of order $2+ε$ are finite for some $ε>0$; otherwise, we say that it is $heavy$-$tailed$. We study queueing networks that operate under the Max-Weight scheduling policy, for the case where some queues receive heavy-tailed and some receive light-tailed traffic. Queues with light-tailed arrivals are often delay stable (that is, expected queue sizes are uniformly bounded over time) but can also become delay unstable because of resource-sharing with other queues that receive heavy-tailed arrivals. Within this context, and for any given "t…
In network theory, Pearson's correlation coefficients are most commonly used to measure the degree assortativity of a network. We investigate the behavior of these coefficients in the setting of directed networks with heavy-tailed degree sequences. We prove that for graphs where the in- and out-degree sequences satisfy a power law with realistic parameters, Pearson's correlation coefficients converge to a non-negative number in the infinite network size limit. We propose alternative measures for degree-degree dependencies in directed networks based on Spearman's rho and Kendall's tau. Using ex…
We develop accurate approximations of the delay distribution of the MArP/G/1 queue that cap- ture the exact tail behavior and provide bounded relative errors. Motivated by statistical analysis, we consider the service times as a mixture of a phase-type and a heavy-tailed distribution. With the aid of perturbation analysis, we derive corrected phase-type approximations as a sum of the delay in an MArP/PH/1 queue and a heavy-tailed component depending on the perturbation parameter. We exhibit their performance with numerical examples.
For moving average processes with random coefficients and heavy-tailed innovations that are weakly dependent in the sense of strong mixing and local dependence condition $D'$ we study joint functional convergence of partial sums and maxima. Under the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we derive a functional limit theorem in the space of $\mathbb{R}^{2}$-valued càdlàg functions on $[0, 1]$ with the Skorokhod weak $M_{2}$ topology.
In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the $\ga$-stable regenerative set. We then apply these results to the strip wetting model which is a random walk $S$ constrained above a wall and rewarded or penalized when it hits the strip $[0,\infty) \times [0,a]$ where $a$ is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality.
We consider the problem of packet scheduling in single-hop queueing networks, and analyze the impact of heavy-tailed traffic on the performance of Max-Weight scheduling. As a performance metric we use the delay stability of traffic flows: a traffic flow is delay stable if its expected steady-state delay is finite, and delay unstable otherwise. First, we show that a heavy-tailed traffic flow is delay unstable under any scheduling policy. Then, we focus on the celebrated Max-Weight scheduling policy, and show that a light-tailed flow that conflicts with a heavy-tailed flow is also delay unstable…
We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $α$-stable limit distributions o…
Rate of convergence is studied for a diffusion process on the half line with a non-sticky reflection to a heavy-tailed 1D invariant distribution which density on the half line has a polynomial decay at infinity. Starting from a standard receipt which guarantees some polynomial convergence, it is shown how to construct a new non-degenerate diffusion process on the half line which converges to the same invariant measure exponentially fast uniformly with respect to the initial data.
We generalize our former localization result about the principal Dirichlet eigenvector of the i.i.d. heavy-tailed random conductance Laplacian to the first $k$ eigenvectors. We overcome the complication that the higher eigenvectors have fluctuating signs by invoking the Bauer-Fike theorem to show that the $k$th eigenvector is close to the principal eigenvector of an auxiliary spectral problem.
It is known that large deviations of sums of subexponential random variables are most likely realised by deviations of a single random variable. In this article we give a detailed picture of how subexponential random variables are distributed when a large deviation of their sum is observed.
Heavy-tailed random variables have been used in insurance research to model both loss frequencies and loss severities, with substantially more emphasis on the latter. In the present work, we take a step toward addressing this imbalance by exploring the class of heavy-tailed frequency models formed by continuous mixtures of Negative Binomial and Poisson random variables. We begin by defining the concept of a calibrative family of mixing distributions (each member of which is identifiable from its associated Negative Binomial mixture), and show how to construct such families from only a single m…
We analyze the asymptotics of crossing a high piecewise linear barriers by a renewal compound process with the subexponential jumps. The study is motivated by ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions when the initial reserves of both companies tend to infinity.
