On the tail behavior of a class of multivariate conditionally heteroskedastic processes

Conditions for geometric ergodicity of multivariate autoregressive conditional heteroskedasticity (ARCH) processes, with the so-called BEKK (Baba, Engle, Kraft, and Kroner) parametrization, are considered. We show for a class of BEKK-ARCH processes that the invariant distribution is regularly varying. In order to account for the possibility of different tail indices of the marginals, we consider the notion of vector scaling regular variation (VSRV), closely related to non-standard regular variation. The characterization of the tail behavior of the processes is used for deriving the asymptotic properties of the sample covariance matrices.     

Spectral tail processes and max-stable approximations of multivariate regularly varying time series

A regularly varying time series as introduced in Basrak and Segers (2009) is a (multivariate) time series such that all finite dimensional distributions are multivariate regularly varying. The extremal behavior of such a process can then be described by the index of regular variation and the so-called spectral tail process, which is the limiting distribution of the rescaled process, given an extreme event at time 0. As shown in Basrak and Segers (2009), the stationarity of the underlying time series implies a certain structure of the spectral tail process, informally known as the “time change formula”. In this article, we show that on the other hand, every process which satisfies this property is in fact the spectral tail process of an underlying stationary max-stable process. The spectral tail process and the corresponding max-stable process then provide two complementary views on the extremal behavior of a multivariate regularly varying stationary time series.     

Branching random walks,stable point processes and regular variation

Using the theory of regular variation, we give a sufficient condition for a point process to be in the superposition domain of attraction of a strictly stable point process. This sufficient condition is used to obtain the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. Because of heavy tails of the step size distribution, we can invoke a one large jump principle at the level of point processes to give an explicit representation of the limiting point process. As a consequence, we extend the main result of Durrett (1983) and verify that two related predictions of Brunet and Derrida (2011) remain valid for this model.     

Operator Tail Dependence of Copulas

A notion of tail dependence based on operator regular variation is introduced for copulas, and the standard tail dependence used in the copula literature is included as a special case. The non-standard tail dependence with marginal power scaling functions having possibly distinct tail indexes is investigated in detail. We show that the copulas with operator tail dependence, incorporated with regularly varying univariate margins, give rise to a rich class of the non-standard multivariate regularly varying distributions. We also show that under some mild conditions, the copula of a non-standard multivariate regularly varying distribution has the standard tail dependence of order 1. Some illustrative examples are given.     

Tail dependence for elliptically contoured distributions

The relationship between the theory of elliptically contoured distributions and the concept of tail dependence is investigated. We show that bivariate elliptical distributions possess the so-called tail dependence property if the tail of their generating random variable is regularly varying, and we give a necessary condition for tail dependence which is somewhat weaker than regular variation of the latter tail. In addition, we discuss the tail dependence property for some well-known examples of elliptical distributions, such as the multivariate normal, t, logistic, and Bessel distributions.     

The tail empirical process for long memory stochastic volatility sequences

This paper describes the limiting behaviour of tail empirical processes associated with long memory stochastic volatility models. We show that such a process has dichotomous behaviour, according to an interplay between the Hurst parameter and the tail index. On the other hand, the tail empirical process with random levels never suffers from long memory. This is very desirable from a practical point of view, since such a process may be used to construct the Hill estimator of the tail index. To prove our results we need to establish new results for regularly varying distributions, which may be of independent interest.     

Refined discrete regular variation and its applications

We introduce a new class of the so‐called regularly varying sequences with respect to τ and state its properties. This class, on one hand, generalizes regularly varying sequences. On the other hand, it refines them and makes it possible to do a more sophisticated analysis in applications. We show a close connection with regular variation on time scales; thanks to this relation, we can use the existing theory on time scales to develop discrete regular variation with respect to τ. We reveal also a connection with generalized regularly varying functions. As an application, we study asymptotic behavior of solutions to linear difference equations; we obtain generalization and extension of known results. The theory also yields, in some way, a new view on the tests for convergence and divergence of series; we establish the statement that generalizes Raabe test and Bertrand test.     

Topological regular variation: II. The fundamental theorems

This paper investigates fundamental theorems of regular variation (Uniform Convergence, Representation, and Characterization Theorems) some of which, in the classical setting of regular variation in R, rely in an essential way on the additive semigroup of natural numbers N (e.g. de Bruijn's Representation Theorem for regularly varying functions). Other such results include Goldie's direct proof of the Uniform Convergence Theorem and Seneta's version of Kendall's theorem connecting sequential definitions of regular variation with their continuous counterparts (for which see Bingham and Ostaszewski (2010) [13]). We show how to interpret these in the topological group setting established in Bingham and Ostaszewski (2010) [12] as connecting N-flow and R-flow versions of regular variation, and in so doing generalize these theorems to Rd. We also prove a flow version of the classical Characterization Theorem of regular variation.     

Multivariate Markov-switching ARMA processes with regularly varying noise

The tail behaviour of stationary R^d-valued Markov-switching ARMA (MS-ARMA) processes driven by a regularly varying noise is analysed. It is shown that under appropriate summability conditions the MS-ARMA process is again regularly varying as a sequence. Moreover, it is established that these summability conditions are satisfied if the sum of the norms of the autoregressive parameters is less than one for all possible values of the parameter chain, which leads to feasible sufficient conditions.Our results complement in particular those of Saporta [Tail of the stationary solution of the stochastic equation Y_n+1=a_nY_n+b_n with Markovian coefficients, Stochastic Process. Appl. 115 (2005) 1954-1978.] where regularly varying tails of one-dimensional MS-AR(1) processes coming from consecutive large values of the parameter chain were studied.     

Functional weak convergence of partial maxima processes

For a strictly stationary sequence of nonnegative regularly varying random variables (X _n ) we study functional weak convergence of partial maxima processes \(M_{n}(t) = \bigvee _{i=1}^{\lfloor nt \rfloor }X_{i},\,t \in [0,1]\) in the space D[0, 1] with the Skorohod J ₁ topology. Under the strong mixing condition, we give sufficient conditions for such convergence when clustering of large values do not occur. We apply this result to stochastic volatility processes. Further we give conditions under which the regular variation property is a necessary condition for J ₁ and M ₁ functional convergences in the case of weak dependence. We also prove that strong mixing implies the so-called Condition \(\mathcal {A}(a_{n})\) with the time component.     

Poisson limits for U-statistics

We study Poisson limits for U-statistics with non-negative kernels. The limit theory is derived from the Poisson convergence of suitable point processes of U-statistics structure. We apply these results to derive infinite variance stable limits for U-statistics with a regularly varying kernel and to determine the index of regular variation of the left tail of the kernel. The latter is known as correlation dimension. We use the point process convergence to study the asymptotic behavior of some standard estimators of this dimension.     

批到达M/G/1重试排队的队长的尾渐近

用随机分解法研究成批到达服务时间为次指数分布的重试排队中队长的尾行为,得到了该系统与其相应的标准排队系统队长尾分布的关系;对次指数尾,结果也能用于正则变化尾,进而得到正则变化尾渐近．     

On the regular variation of ratios of jointly Fréchet random variables

We provide a necessary and sufficient condition for the ratio of two jointly α-Fréchet random variables to be regularly varying. This condition is based on the spectral representation of the joint distribution and is easy to check in practice. Our result motivates the notion of the ratio tail index, which quantifies dependence features that are not characterized by the tail dependence index. As an application, we derive the asymptotic behavior of the quotient correlation coefficient proposed in Zhang (Ann Stat 36(2):1007–1030, 2008) in the dependent case. Our result also serves as an example of a new type of regular variation of products, different from the ones investigated by Maulik et al (J Appl Probab 39(4):671–699, 2002).     

Asymptotics in a time-dependent renewal risk model with stochastic return

In this paper we study the asymptotic tail behavior for a non-standard renewal risk model with a dependence structure and stochastic return. An insurance company is allowed to invest in financial assets such as risk-free bonds and risky stocks, and the price process of its portfolio is described by a geometric Lévy process. By restricting the claim-size distribution to the class of extended regular variation (ERV) and imposing a constraint on the Lévy process in terms of its Laplace exponent, we obtain for the tail probability of the stochastic present value of aggregate claims a precise asymptotic formula, which holds uniformly for all time horizons. We further prove that the corresponding ruin probability also satisfies the same asymptotic formula.     

Testing for small bias of tail index estimators

We determine the joint asymptotic normality of kernel and weighted least-squares estimators of the upper tail index of a regularly varying distribution when each estimator is a bivariate function of two parameters: the tuning parameter is motivated by possible underlying second-order behavior in regular variation, while no such behavior is assumed, and the fraction parameter determines that upper portion of the sample on which the estimator is based. Under the hypothesis that the scaled asymptotic biases of the estimators vanish uniformly in the parameter points considered, these results imply joint asymptotic normality for deviations of ratios of the estimators from 1, which in turn yield asymptotic chi-square tests for checking the small-bias hypothesis, equivalent to the constructibility of asymptotic confidence intervals. The test procedure suggests adaptive choices of the tuning and fraction parameters: data-driven (t)estimators.     

Tail Risk of Multivariate Regular Variation

Tail risk refers to the risk associated with extreme values and is often affected by extremal dependence among multivariate extremes. Multivariate tail risk, as measured by a coherent risk measure of tail conditional expectation, is analyzed for multivariate regularly varying distributions. Asymptotic expressions for tail risk are established in terms of the intensity measure that characterizes multivariate regular variation. Tractable bounds for tail risk are derived in terms of the tail dependence function that describes extremal dependence. Various examples involving Archimedean copulas are presented to illustrate the results and quality of the bounds.     

Optimal impulse and regular control strategies for proportional reinsurance problem

We formulate a stochastic control problem on proportional reinsurance that includes impulse and regular control strategies. For the first time we combine impulse control with regular control, and derive the expected total discount pay-out (return function) from present to bankruptcy. By relying on both stochastic calculus and the classical theory of impulse and regular controls, we state a set of sufficient conditions for its solution in terms of optimal return function. Moreover, we also derive its explicit form and corresponding impulse and regular control strategies.     

Second order tail asymptotics for the sum of dependent, tail-independent regularly varying risks

In this paper we consider dependent random variables with common regularly varying marginal distribution. Under the assumption that these random variables are tail-independent, it is well known that the tail of the sum behaves like in the independence case. Under some conditions on the marginal distributions and the dependence structure (including Gaussian copula’s and certain Archimedean copulas) we provide the second-order asymptotic behavior of the tail of the sum.     

Long memory in a linear stochastic Volterra differential equation

In this paper we consider a linear stochastic Volterra equation which has a stationary solution. We show that when the kernel of the fundamental solution is regularly varying at infinity with a log-convex tail integral, then the autocovariance function of the stationary solution is also regularly varying at infinity and its exact pointwise rate of decay can be determined. Moreover, it can be shown that this stationary process has either long memory in the sense that the autocovariance function is not integrable over the reals or is subexponential. Under certain conditions upon the kernel, even arbitrarily slow decay rates of the autocovariance function can be achieved. Analogous results are obtained for the corresponding discrete equation.     

Extremal solutions to a system of n nonlinear differential equations and regularly varying functions

The strongly increasing and strongly decreasing solutions to a system of n nonlinear first order equations are here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We establish conditions under which such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish asymptotic representations. Several applications of the main results are given, involving n‐th order nonlinear differential equations, equations with a generalized ?‐Laplacian, and nonlinear partial differential systems.