The notion of a phantom distribution function (phdf) was introduced by O'Brien (Ann. Probab. 15, 281–292 (1987)). We show that the existence of a phdf is a quite common phenomenon for stationary weakly dependent sequences. It is proved that any α-mixing stationary sequence with continuous marginals admits a continuous phdf. Sufficient conditions are given for stationary sequences exhibiting weak dependence, what allows the use of attractive models beyond mixing. The case of discontinuous marginals is also discussed for α-mixing. Special attention is paid to examples of processes which admit a continuous phantom distribution function while their extremal index is zero. We show that Asmussen (Ann. Appl. Probab. 8, 354–374 1998) and Roberts et al. (Extremes. 9, 213–229 2006) provide natural examples of such processes. We also construct a non-ergodic stationary process of this type.

Considering extreme quantiles is a popular way to understand the tail of a distribution. While they have been extensively studied for univariate distributions, much less has been done for multivariate ones, primarily because there is no universally accepted definition of what a multivariate quantile or a multivariate distribution tail should be. In this paper, we focus on extreme geometric quantiles. In Girard and Stupfler (2015) Intriguing properties of extreme geometric quantiles, their asymptotics are established, both in direction and magnitude, under suitable integrability conditions, when the norm of the associated index vector tends to one. In this paper, we study extreme geometric quantiles when the integrability conditions are not fulfilled, in a framework of regular variation.

We obtain a number of new general properties, related to the closedness of the class of long-tailed distributions under convolutions, that are of interest themselves and may be applied in many models that deal with “plus” and/or “max” operations on heavy-tailed random variables. We analyse the closedness property under convolution roots for these distributions. Namely, we introduce two classes of heavy-tailed distributions that are not long-tailed and study their properties. These examples help to provide further insights and, in particular, to show that the properties to be both long-tailed and so-called “generalised subexponential” are not preserved under the convolution roots. This leads to a negative answer to a conjecture of Embrechts and Goldie (J. Austral. Math. Soc. (Ser. A) 29, 243–256 1980, Stoch. Process. Appl. 13, 263–278 1982) for the class of long-tailed and generalised subexponential distributions. In particular, our examples show that the following is possible: an infinitely divisible distribution belongs to both classes, while its Levy measure is neither long-tailed nor generalised subexponential.

In this paper, we study the asymptotic distribution of the number of exceedances of multiplicative factor models under the rare-event condition that the probability the number of exceedances is positive tends to zero. We show that the asymptotic conditional distribution (given that there is at least one exceedance), as well as the condition such that the probability of the conditional event tends to zero, depend on the distribution of the multiplicative disturbances. When the distribution of the disturbances is not a heavy-tailed distribution, it is necessary to normalize the number of exceedances to get a non-degenerate asymptotic distribution.

Let X1, … , Xn be independent nonnegative random variables with respective survival functions \(\overline {F}_{1}, \ldots , \overline {F}_{n}\), and let Θ1, … , Θn be (not necessarily independent) nonnegative random variables, independent of X1, … , Xn, satisfying certain moment conditions. This paper consists of two parts. In the first part, we investigate second-order expansions of 𝗣 \( \left ({\sum }^{n}_{i=1} X_{i}>t\right )\) as \(t\to {\infty }\) under the assumption that the \(\overline {F}_{i}\) are of second-order regular variation (2RV) with the same first-order index but with different second-order indexes. In the second part, under the assumption that the \(\overline {F}_{1}=\cdots =\overline {F}_{n}\) have 2RV tails, second-order expansions of tail probabilities of the randomly weighted sum \({\sum }^{n}_{i=1} {\Theta }_{i} X_{i}\) are studied. The closure property of 2RV under randomly weighted sum is also discussed. The main results in this paper generalize and strengthen several known results in the literature.

Abel-Tauberian theorems relate power law behavior of distributions and their transforms. We formulate and prove a multivariate version for non-standard regularly varying measures on \(\mathbb {R}_{+}^{p}\) and then apply it to prove that the joint distribution of in- and out-degree in a directed edge preferential attachment model has jointly regularly varying tails.

For a centered d-dimensional Gaussian random vector ξ = (ξ 1, … , ξ d ) and a homogeneous function h : ℝd → ℝ we derive asymptotic expansions for the tail of the Gaussian chaos h(ξ) given the function h is sufficiently smooth. Three challenging instances of the Gaussian chaos are the determinant of a Gaussian matrix, the Gaussian orthogonal ensemble and the diameter of random Gaussian clouds. Using a direct probabilistic asymptotic method, we investigate both the asymptotic behaviour of the tail distribution of h(ξ) and its density at infinity and then discuss possible extensions for some general ξ with polar representation.

We consider heavy tailed time series whose finite-dimensional distributions are extremally independent in the sense that extremely large values cannot be observed consecutively. This calls for methods beyond the classical multivariate extreme value theory which is convenient only for extremally dependent multivariate distributions. We use the Conditional Extreme Value approach to study the effect of an extreme value at time zero on the future of the time series. In formal terms, we study the limiting conditional distribution of future observations given an extreme value at time zero. To this purpose, we introduce conditional scaling functions and conditional scaling exponents. We compute these quantities for a variety of models, including Markov chains, exponential autoregressive models, stochastic volatility models with heavy tailed innovations or volatilities.

Extremal dependence of the losses in a portfolio is one of the most important features that should be accounted for when estimating Value-at-Risk (VaR) at high levels. Multivariate extreme value theory provides a principled framework for the modeling and estimation of extremal dependence. In practice, however, this involves dealing with a challenging infinite dimensional parameter such as the spectral measure. Here, following recent developments in Schlather and Tawn (Extremes, 5(1) 87–102; 2002), Molchanov (Extremes, 11, 235–259; 2008), and Strokorb and Schlather (2013), we propose to represent extremal dependence of a multivariate portfolio via the so–called Tawn–Molchanov (TM) model, which is finite dimensional. Every max–stable random vector X can be associated with a TM max-stable vector Y = TM(X) so that the extremal coefficients of X and Ymatch and at the same time Y stochastically dominates X in the lower orthant order. This result readily yields an optimal upper bound on the value-at-risk \(\text {VaR}_{\alpha }(\mathbf {X}^{\vee })\) of the maximum portfolio loss \(\mathbf {X}^{\vee }:=\max _{j=1,\ldots ,d}X_{j}\). We develop a statistical methodology for estimating TM models from data and illustrate the resulting upper bounds on \(\text {VaR}_{\alpha }(\mathbf {X}^{\vee })\) with simulations and real data. Fitting TM models to portfolio data may be of independent practical interest, since their coefficients provide a qualitative picture of the degree and nature of diversification to extreme shocks.