Subexponential Tails Research Articles

Exponential growth of random determinants beyond invariance

We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with covariance profiles, Wigner matrices and covariance matrices with subexponential tails, Erd\H{o}s-R\'enyi and $d$-regular graphs for any polynomial sparsity parameter, and non-mean-field random matrix models, such as random band matrices for any polynomial bandwidth. The proof builds on recent tools, including the theory of the Matrix Dyson Equation as developed in [Ajanki, Erd\H{o}s, Kr\"uger 2019]. We use these asymptotics as an important input to identify the complexity of classes of Gaussian random landscapes in our companion papers [Ben Arous, Bourgade, McKenna 2021; McKenna 2021].

Extremal clustering under moderate long range dependence and moderately heavy tails

We study clustering of the extremes in a stationary sequence with subexponential tails in the maximum domain of attraction of the Gumbel distribution. We obtain functional limit theorems in the space D[0,∞) and in the space of random sup-measures. The limits have the Gumbel distribution if the memory is only moderately long. However, as our results demonstrate rather strikingly, the “heuristic of a single big jump” could fail even in a moderately long range dependence setting. As the tails become lighter, the extremal behavior of a stationary process may depend on multiple large values of the driving noise.

On the finiteness and tails of perpetuities under a Lamperti–Kiu MAP

AbstractConsider a Lamperti–Kiu Markov additive process $(J, \xi)$ on $\{+, -\}\times\mathbb R\cup \{-\infty\}$, where J is the modulating Markov chain component. First we study the finiteness of the exponential functional and then consider its moments and tail asymptotics under Cramér’s condition. In the strong subexponential case we determine the subexponential tails of the exponential functional under some further assumptions.

Asymptotics for a time-dependent renewal risk model with subexponential main claims and delayed claims

This paper considers a nonstandard renewal risk model with constant force of interest, in which each main claim induces a delayed claim. Assume that the main claim sizes and the inter-arrival times obey a dependence structure, and so do the delayed claim sizes and the arrival times. Supposing that the distributions of the main and delayed claim sizes both have subexponential tails, asymptotic estimate for the finite-time ruin probability of this risk model is obtained, as the initial surplus tends to infinity.

Diffusions interacting through a random matrix: universality via stochastic Taylor expansion

Consider (X_{i}(t)) solving a system of N stochastic differential equations interacting through a random matrix {mathbf {J}} = (J_{ij}) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of (X_i(t)), initialized from some mu independent of {mathbf {J}}, are universal, i.e., only depend on the choice of the distribution mathbf {J} through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.

Asymptotic normality for random simplices and convex bodies in high dimensions

Central limit theorems for the log-volume of a class of random convex bodies in R n \mathbb {R}^n are obtained in the high-dimensional regime, that is, as n → ∞ n\to \infty . In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are assumed to be independent and identically distributed with subexponential tails. In addition, asymptotic normality is also established for random convex bodies (including random simplices pinned at the origin) when the spanning vectors are distributed according to a radially symmetric probability measure on the n n -dimensional ℓ p \ell _p -ball. In particular, this includes the cone and the uniform probability measure.

The bias of isotonic regression.

We study the bias of the isotonic regression estimator. While there is extensive work characterizing the mean squared error of the isotonic regression estimator, relatively little is known about the bias. In this paper, we provide a sharp characterization, proving that the bias scales as O(n -β/3) up to log factors, where 1 ≤ β ≤ 2 is the exponent corresponding to Hölder smoothness of the underlying mean. Importantly, this result only requires a strictly monotone mean and that the noise distribution has subexponential tails, without relying on symmetric noise or other restrictive assumptions.

A New Class of Robust Observation-Driven Models

This paper introduces a new class of observation-driven models, including score models as a special case. This new class inherits and extends the basic ideas behind the development of score models and addresses a number of unsolved issues in the score literature. In particular, the new class of models (i) allows QML estimation of static parameters, (ii) allows the production of leverage effects in the presence of negative outliers, (iii) allows update asymmetry and asymmetric forecast loss functions in the presence of symmetric or skewed innovations, and (iii) achieves out-of-sample outlier robustness in the presence of sub-exponential tails. We establish the asymptotic properties of the QLE, QMLE, and MLE as well as likelihood ratio and Lagrange multiplier test statistics. The finite sample properties are studied by means of an extensive Monte Carlo study. Finally, we show the empirical relevance of this new class of models on real data.

Most trees are short and fat

This work proves new probability bounds relating to the height, width, and size of Galton–Watson trees. For example, if T is any Galton–Watson tree, and H, W, and |T| are the height, width, and size of T, respectively, then H / W has sub-exponential tails and $$H/|T|^{1/2}$$ has sub-Gaussian tails. Although our methods apply without any assumptions on the offspring distribution, when information is provided about the distribution the method can be adapted accordingly, and always seems to yield tight bounds.

Markov dependence in renewal equations and random sums with heavy tails

ABSTRACTThe Markov renewal equationis considered in the subcritical case where the matrix of total masses of the Fij has spectral radius strictly less than one, and the asymptotics of the Zi(x) is found in the heavy-tailed case involving a local subexponential assumption on the Fij. Three cases occur according to the balance between the zi(x) and the tails of the Fij. A crucial step in the analysis is obtaining multivariate and local versions of a lemma due to Kesten on domination of subexponential tails. These also lead to various results on tail asymptotics of sums of a random number of heavy-tailed random variables in models which are more general than in the literature.

On subexponential tails for the maxima of negatively driven compound renewal and Lévy processes

We study subexponential tail asymptotics for the distribution of the maximum Mt≔supu∈[0,t]Xu of a process Xt with negative drift for the entire range of t>0. We consider compound renewal processes with linear drift and Lévy processes. For both processes we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes with drift particularly includes the Cramér–Lundberg renewal risk process.

Asymptotic ruin probabilities for a bidimensional renewal risk model

This paper studies a bidimensional renewal risk model with constant force of interest and heavy-tailed claims. We extend the existing models through allowing arbitrary dependences between each pair of inter-arrival times of the two kinds of insurance claims. A precise asymptotic formula for the finite-time ruin probability is obtained when the claims have subexponential tails. Further restricting the claim-size distributions within the class of extended regular variation, we derive the corresponding asymptotic formula for the infinite-time ruin probability.

Universality of covariance matrices

In this paper we prove the universality of covariance matrices of the form $H_{N\times N}={X}^{\dagger}X$ where $X$ is an ${M\times N}$ rectangular matrix with independent real valued entries $x_{ij}$ satisfying $\mathbb{E}x_{ij}=0$ and $\mathbb{E}x^{2}_{ij}={\frac{1}{M}}$, $N,M\to \infty$. Furthermore it is assumed that these entries have sub-exponential tails or sufficiently high number of moments. We will study the asymptotics in the regime $N/M=d_{N}\in(0,\infty),\lim_{N\to\infty}d_{N}\neq0,\infty$. Our main result is the edge universality of the sample covariance matrix at both edges of the spectrum. In the case $\lim_{N\to\infty}d_{N}=1$, we only focus on the largest eigenvalue. Our proof is based on a novel version of the Green function comparison theorem for data matrices with dependent entries. En route to proving edge universality, we establish that the Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by the Marcenko–Pastur law uniformly up to the edges of the spectrum with an error of order $(N\eta)^{-1}$ where $\eta$ is the imaginary part of the spectral parameter in the Stieltjes transform. Combining these results with existing techniques we also show bulk universality of covariance matrices. All our results hold for both real and complex valued entries.

Tachyonic quantum densities of relativistic electron plasmas: Cherenkov spectra of γ-ray pulsars

Abstract Tachyonic Cherenkov radiation in second quantization can explain the subexponential spectral tails of GeV γ-ray pulsars (Crab pulsar, PSR J1836+5925, PSR J0007+7303, PSR J2021+4026) recently observed with the Fermi-LAT, VERITAS and MAGIC telescopes. The radiation is emitted by a thermal ultra-relativistic electron plasma. The Cherenkov effect is derived from a Maxwell–Proca field with negative mass-square in a dispersive spacetime. The frequency variation of the tachyon mass results in exp ( − β ˆ ω 1 − ρ ) attenuation of the asymptotic Cherenkov energy flux, where β ˆ is a decay constant related to the electron temperature and ρ is the frequency scaling exponent of the tachyon mass. An exponent in the range 0 ρ 1 can reproduce the observed subexponential decay of the energy flux. For the Crab pulsar, we find ρ = 0.81 ± 0.02 , inferred from the substantially weaker-than-exponential decay of its spectral tail measured by MAGIC over an extended energy range. The scaling exponent ρ determines whether the group velocity of the tachyonic γ-rays is sub- or superluminal.

On higher order isotropy conditions and lower bounds for sparse quadratic forms

This study aims at contributing to lower bounds for empirical compatibility constants or empirical restricted eigenvalues. This is of importance in compressed sensing and theory for $\ell_{1}$-regularized estimators. Let $X$ be an $n\times p$ data matrix with rows being independent copies of a $p$-dimensional random variable. Let $\hat{\Sigma}:=X^{T}X/n$ be the inner product matrix. We show that the quadratic forms $u^{T}\hat{\Sigma}u$ are lower bounded by a value converging to one, uniformly over the set of vectors $u$ with $u^{T}\Sigma_{0}u$ equal to one and $\ell_{1}$-norm at most $M$. Here $\Sigma_{0}:=\mathbb{E} \hat{\Sigma}$ is the theoretical inner product matrix, which we assume to exist. The constant $M$ is required to be of small order $\sqrt{n/\log p}$. We assume moreover $m$-th order isotropy for some $m>2$ and sub-exponential tails or moments up to order $\log p$ for the entries in $X$. As a consequence, we obtain convergence of the empirical compatibility constant to its theoretical counterpart, and similarly for the empirical restricted eigenvalue. If the data matrix $X$ is first normalized so that its columns all have equal length we obtain lower bounds assuming only isotropy and no further moment conditions on its entries. The isotropy condition is shown to hold for certain martingale situations.

Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims

In this paper, we obtain the finite-horizon and infinite-horizon ruin probability asymptotics for risk processes with claims of subexponential tails for non-stationary arrival processes that satisfy a large deviation principle. As a result, the arrival process can be dependent, non-stationary and non-renewal. We give three examples of non-stationary and non-renewal point processes: Hawkes process, Cox process with shot noise intensity and self-correcting point process. We also show some aggregate claims results for these three examples.

Second order corrections for the limits of normalized ruin times in the presence of heavy tails

In this paper we consider a compound Poisson risk model with regularly varying claim sizes. For this model in [1] an asymptotic formula for the finite time ruin probability is provided when the time is scaled by the mean excess function. In this paper we derive the rate of convergence for this finite time ruin probability when the claims have a finite second moment. [1] S. Asmussen and C. Kl\uppelberg. Large deviations results for subexponential tails, with applications to insurance risk. Stochastic Process. Appl., 64(1):103-125, 1996.

Large deviations for random sums of differences between two sequences of random variables with applications to risk theory

This paper investigates some precise large deviations for the random sums of the differences between two sequences of independent and identically distributed random variables, where the minuend random variables have subexponential tails, and the subtrahend random variables have finite second moments. As applications to risk theory, the customer-arrival-based insurance risk model is considered, and some uniform asymptotics for the ruin probabilities of an insurance company are derived as the number of customers or the time tends to infinity. MSC: 60F10; 62E20; 62P05

New examples of heavy-tailed O-subexponential distributions and related closure properties

Let L and S denote the classes of distributions with long tails and subexponential tails respectively. Let OS denote the class of distributions with O-subexponential tails, which means the distributions with the tails having the same order as the tails of their 2-fold convolutions. In this paper, we first construct a family of distributions without finite means in L ∩ OS ∖ S . Next some distributions in L ∩ OS ∖ S , which possess finite means or even finite higher moments, are also constructed. In connection with this, we prove that the class OS is closed under minimization of random variables. However, it is not closed under maximization of random variables.

Uniform estimates for the finite-time ruin probability in the dependent renewal risk model

This paper investigates the finite-time ruin probability in the dependent renewal risk model, where the claim sizes are independent and identically distributed random variables with strongly subexponential tails, and the interarrival times are negatively dependent. We establish an asymptotic estimate, which holds uniformly for the time horizon varying in the positive half line.