Tail Asymptotics Research Articles

Stable Lévy motion with inverse Gaussian subordinator

In this paper we study the stable Lévy motion subordinated by the so-called inverse Gaussian process. This process extends the well known normal inverse Gaussian (NIG) process introduced by Barndorff-Nielsen, which arises by subordinating ordinary Brownian motion (with drift) with inverse Gaussian process. The NIG process found many interesting applications, especially in financial data description. We discuss here the main features of the introduced subordinated process, such as distributional properties, existence of fractional order moments and asymptotic tail behavior. We show the connection of the process with continuous time random walk. Further, the governing fractional partial differential equations for the probability density function is also obtained. Moreover, we discuss the asymptotic distribution of sample mean square displacement, the main tool in detection of anomalous diffusion phenomena (Metzler et al., 2014). In order to apply the stable Lévy motion time-changed by inverse Gaussian subordinator we propose a step-by-step procedure of parameters estimation. At the end, we show how the examined process can be useful to model financial time series.

A unified approach for large queue asymptotics in a heterogeneous multiserver queue

Abstract We are interested in a large queue in a GI/G/k queue with heterogeneous servers. For this, we consider tail asymptotics and weak limit approximations for the stationary distribution of its queue length process in continuous time under a stability condition. Here, two weak limit approximations are considered. One is when the variances of the interarrival and/or service times are bounded, and the other is when they become large. Both require a heavy-traffic condition. Tail asymptotics and heavy-traffic approximations have been separately studied in the literature. We develop a unified approach based on a martingale produced by a good test function for a Markov process to answer both problems.

Auto-correlation function and frequency spectrum due to a super-position of uncorrelated exponential pulses

The auto-correlation function and the frequency power spectral density due to a super-position of uncorrelated exponential pulses are considered. These are shown to be independent of the degree of pulse overlap and thereby the intermittency of the stochastic process. For constant pulse duration and a one-sided exponential pulse shape, the power spectral density has a Lorentzian shape which is flat for low frequencies and a power law at high frequencies. The algebraic tail is demonstrated to result from the discontinuity in the pulse function. For a strongly asymmetric two-sided exponential pulse shape, the frequency spectrum is a broken power law with two scaling regions. In the case of a symmetric pulse shape, the power spectral density is the square of a Lorentzian function. The steep algebraic tail at high frequencies in these cases is demonstrated to follow from the discontinuity in the derivative of the pulse function. A random distribution of pulse durations is shown to result in apparently longer correlation times but has no influence on the asymptotic power law tail of the frequency spectrum. The effect of additional random noise is also discussed, leading to a flat spectrum for high frequencies. The probability density function for the fluctuations is shown to be independent of the distribution of pulse durations. The predictions of this model describe the variety of auto-correlation functions and power spectral densities reported from experimental measurements in the scrape-off layer of magnetically confined plasmas.

The interaction of finite‐width reconnection exhaust jets with a dipolar magnetic field configuration

AbstractThe structure and properties of earthward propagating exhaust jets produced by a magnetic reconnection source region localized in the out‐of‐plane direction are investigated with 3‐D particle‐in‐cell simulations that incorporate a realistic plasma sheet configuration in which a dipole magnetic field region is connected to an asymptotic tail equilibrium with finite Bz. The jet is found to separate initially into two segments of width ∼10–15di each (di is the ion inertia length). The dawnward segment moves ahead of the duskward one and continues to intensify, while the latter is buffeted by the return flows generated by the former and stagnates. The leading front develops the characteristic structure of a sharp Bz increase and density drop on a 1–2di scale. The current associated with this Bz increase is carried mainly by the electrons, and the jet front is the site of a Region 1 sense field‐aligned current system. The ions develop a moderate temperature anisotropy with Ti||>Ti⊥ ahead of the front, while the electron temperature remains isotropic. The ion heating behind the front is quasi‐adiabatic, while the nonthermal electron tail is enhanced out to the limit of the simulation spectrum (∼15Eth). The ion velocity distribution function behind the front exhibits a strong parallel phase space enhancement at speeds ∼0.5VA associated with the earthward flow, while closer to the dipole region there is a drop out in parallel phase space at speeds of ∼1.5–2.5VA. The electron distribution is isotropic with an enhanced central plateau extending out to ∼4VA.

Tail dependence analysis of stock markets using extreme value theory

ABSTRACTFinancial risk modelling frequently uses the assumption of a normal distribution when considering the return series which is inefficient if the data is not normally distributed or if it exhibits extreme tails. Estimation of tail dependence between financial assets plays a vital role in various aspects of financial risk modelling including portfolio theory and hedging amongst applications. Extreme Value Theory (EVT) provides well established methods for considering univariate and multivariate tail distributions which are useful for forecasting financial risk or modelling the tail dependence of risky assets. The empirical analysis in this article uses nonparametric measures based on bivariate EVT to investigate asymptotic dependence and estimate the degree of tail dependence of the ASX-All Ordinaries daily returns with four other international markets, viz., the S&P-500, Nikkei-225, DAX-30 and Heng-Seng for both extreme right and left tails of the return distribution. It is investigated whether the asymptotic dependence between these markets is related to the heteroscedasticity present in the logarithmic return series using GARCH filters. The empirical evidence shows that the asymptotic extreme tail dependence between stock markets does not necessarily exist and rather can be associated with the heteroscedasticity present in the financial time series of the various stock markets.

Stationary analysis of the “Shortest Queue First” service policy: The asymmetric case

ABSTRACTThe Shortest Queue First service discipline is analyzed for two asymmetric queues. For Poisson job arrivals and exponentially distributed service times, we first show that the Laplace transform of workloads in each queue is related to the solution M of a two-dimensional functional equationwith given matrices Q1, Q2 and vector L, where each function h1 and h2 is algebraically defined on a rational cubic curve. The solution M is then represented by a series expansion involving the semi-group generated by the pair of functions h1 and h2. This series expansion together with analytic continuation properties of the solution M enables us to calculate the empty queue probabilities, the mean queue occupancies and to characterize the tail behavior of the workload distribution at each queue.Regarding the application to congestion control in telecommunication networks, our analysis shows that, without explicit marking, the SQF discipline yields a good performance in terms of mean waiting time for flows with a small contribution to the global load; this discipline may, however, have adverse effects for very large values of the waiting time, with tail asymptotics similar to that for the lowest priority flow of an HoL system.

Topological defects with power-law tails

We study interactions of kinks and antikinks of the (1 + 1)-dimensional ϕ8 model. In this model, there are kinks with mixed tail asymptotics: power-law behavior at one infinity versus exponential decay towards the other. We show that if a kink and an antikink face each other in way such that their power-law tails determine the kink-antikink interaction, then the force of their interaction decays slowly, as some negative power of distance between them. We estimate the force numerically using the collective coordinate approximation, and analytically via Manton’s method (making use of formulas derived for the kink and antikink tail asymptotics).

Survival asymptotics for branching random walks in IID environments

We first study a model, introduced recently in [4], of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no ‘obstacle’ placed there. The obstacles appear at each site with probability $p\in [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical. Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $\mathbb P_p$-measure, as $n\to \infty $, $P^{\omega }(S_n)\sim 2/(qn)$ in the critical case, while this probability is asymptotically stretched exponential in the subcritical case. Hence, the model exhibits ‘self-averaging’ in the critical case but not in the subcritical one. I.e., in the first case, the asymptotic tail behavior is the same as in a ‘toy model’ where space is removed, while in the second, the spatial survival probability is larger than in the corresponding toy model, suggesting spatial strategies. A spine decomposition of the branching process along with known results on random walks are utilized.

A note on weighted infinite sums of dependent regularly varying tailed random variables

In this paper we study the asymptotic tail behavior of weighted infinite sums of heavy-tailed random variables under a general dependence structure. Our result shows that the regular variation at tails does not only eliminate the given dependence structure, but the aggregate tail probabilities are asymptotically determined by the sums of the individual tail probabilities with positive coefficients. The result will be illustrated with examples.

Extremes of Gaussian random fields with regularly varying dependence structure

Let $X(t), t\in \mathcal{T}$ be a centered Gaussian random field with variance function $\sigma^2(\cdot)$ that attains its maximum at the unique point $t_0\in \mathcal{T}$, and let $M(\mathcal{T}):=\sup_{t\in \mathcal{T}} X(t)$. For $\mathcal{T}$ a compact subset of $\R$, the current literature explains the asymptotic tail behaviour of $M(\mathcal{T})$ under some regularity conditions including that $1- \sigma(t)$ has a polynomial decrease to 0 as $t \to t_0$. In this contribution we consider more general case that $1- \sigma(t)$ is regularly varying at $t_0$. We extend our analysis to random fields defined on some compact $\mathcal{T}\subset \R^2$, deriving the exact tail asymptotics of $M(\mathcal{T})$ for the class of Gaussian random fields with variance and correlation functions being regularly varying at $t_0$. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.

Discrete hierarchy of sizes and performances in the exchange-traded fund universe

Using detailed statistical analyses of the size distribution of a universe of equity exchange-traded funds (ETFs), we discover a discrete hierarchy of sizes, which imprints a log-periodic structure on the probability distribution of ETF sizes that dominates the details of the asymptotic tail. This allows us to propose a classification of the studied universe of ETFs into seven size layers approximately organized according to a multiplicative ratio of 3.5 in their total market capitalization. Introducing a similarity metric generalising the Herfindhal index, we find that the largest ETFs exhibit a significantly stronger intra-layer and inter-layer similarity compared with the smaller ETFs. Comparing the performance across the seven discerned ETF size layers, we find an inverse size effect, namely large ETFs perform significantly better than the small ones both in 2014 and 2015.

A generalization of multivariate Pareto distributions: tail risk measures, divided differences and asymptotics

We consider a multivariate distribution of the form , where the survival function h is a multiply monotonic function of order such that , for all i and for . This generalizes work by Chiragiev and Landsman on completely monotonic survival functions. We show that the considered dependence structure is more flexible in the sense that the correlation coefficient between two components may attain negative values. We demonstrate that the tool of divided difference is very convenient for evaluation of tail risk measures and their allocations. In terms of divided differences, formulas for tail conditional expectation (tce), tail conditional variance and tce-based capital allocation are obtained. We obtain a closed form for the capital allocation of aggregate risk. Special attention is paid to survival functions h that are regularly or rapidly varying.

Part B: Evaluating Concordance Measures via a Tensor Approximation of Generalized Correlated Diffusions

There is now an increasingly large number of proposed concordance measures available to capture, measure and quantify different notions of dependence in stochastic processes. These can include concepts such as multivariate upper negative (positive) dependence, lower negative (positive) dependence, negative (positive) dependence; multivariate negative (positive) quadrant dependence; multivariate association, co-monotinicity, stochastic ordering; regression dependence negative (positive) and extreme dependence, asymptotic tail dependence and intermediate tail dependence; as well as other concepts such as directional dependence. However, evaluation of concordance measures to quantify such types of dependence for different copula models can be challenging. Therefore, we propose a class of new methods that is highly accurate and computationally efficient procedure to evaluate concordance measures for a given copula. In addition, this then allows us to reconstruct maps of concordance measures locally in all regions of the state space for any range of copula parameters. We believe this technique will be a valuable tool for practitioners to understand better the behaviour of copula models.

Tail asymptotics of generalized deflated risks with insurance applications

Let X and S∈(0,1) be two independent risk variables. This paper investigates approximations of generalized deflated risks E{XκI{SX>x}} with a flexible constant κ≥0 under extreme value theory framework. Our findings are illustrated by three applications concerning higher-order tail approximations of deflated risks as well as approximations of the Haezendonck–Goovaerts and expectile risk measures. Numerical analyses show that higher-order approximations obtained in this paper significantly improve lower-order approximations.

Dominant poles and tail asymptotics in the critical Gaussian many-sources regime

The dominant pole approximation (DPA) is a classical analytic method to obtain from a generating function asymptotic estimates for its underlying coefficients. We apply DPA to a discrete queue in a critical many-sources regime, in order to obtain tail asymptotics for the stationary queue length. As it turns out, this regime leads to a clustering of the poles of the generating function, which renders the classical DPA useless, since the dominant pole is not sufficiently dominant. To resolve this, we design a new DPA method, which might also find application in other areas of mathematics, like combinatorics, particularly when Gaussian scalings related to the central limit theorem are involved.

Extremes of α(t)-locally stationary Gaussian processes with non-constant variances

With motivation from [9], in this paper we derive the exact tail asymptotics of α(t)-locally stationary Gaussian processes with non-constant variance functions. We show that some certain variance functions lead to qualitatively new results.

Time inhomogeneity in longest gap and longest run problems

Consider an inhomogeneous Poisson process and let D be the first of its epochs which is followed by a gap of size ℓ>0. We establish a criterion for D<∞ a.s., as well as for D being long-tailed and short-tailed, and obtain logarithmic tail asymptotics in various cases. These results are translated into the discrete time framework of independent non-stationary Bernoulli trials where the analogue of D is the waiting time for the first run of ones of length ℓ. A main motivation comes from computer reliability, where D+ℓ represents the actual execution time of a program or transfer of a file of size ℓ in presence of failures (epochs of the process) which necessitate restart.

Tail dependence for skew Laplace distribution and skew Cauchy distribution

ABSTRACTCoefficient of tail dependence measures the strength of dependence in the tail of a bivariate distribution and it has been found useful in the risk management. In this paper, we derive the upper tail dependence coefficient for a random vector following the skew Laplace distribution and the skew Cauchy distribution, respectively. The result shows that skew Laplace distribution is asymptotically independent in upper tail, however, skew Cauchy distribution has asymptotic upper tail dependence.

Estimating the coefficient of asymptotic tail independence

Many multivariate analyses require the account of extreme events. Correlation is an insufficient measure to quantify tail dependence. The most common tail dependence coefficients are based on the probability of simultaneous exceedances. The coefficient of asymptotic tail independence introduced in Ledford and Tawn (1996) is a bivariate measure often used in the tail modeling of data in finance, environment, insurance, among other fields of applications. It can be estimated as the tail index of the minimum component of a random pair with transformed unit Pareto marginals. The literature regarding the estimation of the tail index is extensive. Semi-parametric inference requires the choice of the number k of the largest order statistics that lead to the best estimate, where there is a tricky trade-off between variance and bias. Many methodologies have been developed to undertake this choice, most of them applied to the Hill estimator (Hill, 1975). We are going to analyze, through simulation, some of these methods within the estimation of the coefficient of asymptotic tail independence. We also compare with a minimum-variance reduced-bias Hill estimator presented in Caeiro et al. (2005). A pure heuristic procedure adapted from Frahm et al. (2005), used in a different context but with a resembling framework, will also be implemented. We will see that some of these simple tools should not be discarded in this context. Our study will be complemented by applications to real datasets.

Extremes of locally stationary chi-square processes with trend

Chi-square processes with trend appear naturally as limiting processes in various statistical models. In this paper we are concerned with the exact tail asymptotics of the supremum taken over (0,1) of a class of locally stationary chi-square processes with particular admissible trends. An important tool for establishing our results is a weak version of Slepian’s lemma for chi-square processes. Some special cases including squared Brownian bridge and Bessel process are discussed.