Extremal behavior of regularly varying stochastic processes

We study a formulation of regular variation for multivariate stochastic processes on the unit interval with sample paths that are almost surely right-continuous with left limits and we provide necessary and sufficient conditions for such stochastic processes to be regularly varying. A version of the Continuous Mapping Theorem is proved that enables the derivation of the tail behavior of rather general mappings of the regularly varying stochastic process. For a wide class of Markov processes with increments satisfying a condition of weak dependence in the tails we obtain simplified sufficient conditions for regular variation. For such processes we show that the possible regular variation limit measures concentrate on step functions with one step, from which we conclude that the extremal behavior of such processes is due to one big jump or an extreme starting point. By combining this result with the Continuous Mapping Theorem, we are able to give explicit results on the tail behavior of various vectors of functionals acting on such processes. Finally, using the Continuous Mapping Theorem we derive the tail behavior of filtered regularly varying Lévy processes.     

Estimating asymptotic dependence functionals in multivariate regularly varying models

This paper deals with semiparametric estimation of the asymptotic portfolio risk factor ?? _?? introduced in [G. Mainik and L. Rüschendorf, On optimal portfolio diversification with respect to extreme risks, Finance Stoch., 14:593?C623, 2010] for multivariate regularly varying random vectors in $ \mathbb{R}_{+}^d $ . The functional ?? _?? depends on the spectral measure ??, the tail index ??, and the vector ?? of portfolio weights. The representation of ?? _?? is extended to characterize the portfolio loss asymptotics for random vectors in ? ^d . The earlier results on uniform strong consistency and uniform asymptotic normality of the estimates of ?? _?? are extended to the general setting, and the regularity assumptions are significantly weakened. Uniform consistency and asymptotic normality are also proved for the estimators of the functional $ \gamma_\xi^{{{1} \left/ {\alpha } \right.}} $ that characterizes the asymptotic behavior of the portfolio loss quantiles. The techniques developed here can also be applied to other dependence functionals.     

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 σ ²(?) 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 ?, the current literature explains the asymptotic tail behaviour of \(M(\mathcal {T})\) under some regularity conditions including that 1 ? σ(t) has a polynomial decrease to 0 as t → t ₀. In this contribution we consider more general case that 1 ? σ(t) is regularly varying at t ₀. We extend our analysis to Gaussian random fields defined on some compact set \(\mathcal {T}\subset \mathbb {R}^{2}\), 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.     

Small time convergence of subordinators with regularly or slowly varying canonical measure

We consider subordinators $X_{\alpha }={\left(X_{\alpha }\left(t\right)\right)}_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator ${\left(S_{\alpha }\left(t\right)\right)}_{t\ge 0}$ (where $\alpha \in (0,1)$); thus, with the property that ${\overline{\Pi }}_{\alpha }$, the tail function of the canonical measure of $X_{\alpha }$, is regularly varying of index $?\alpha \in (?1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $\alpha =0$, when ${\overline{\Pi }}_{\alpha }$ is slowly varying at 0. When $\alpha \in (0,1)$, we show that ${\left(t{\overline{\Pi }}_{\alpha }\left(X_{\alpha }\left(t\right)\right)\right)}^{?1}$ converges in distribution, as $t\downarrow 0$, to the random variable ${\left(S_{\alpha }\left(1\right)\right)}^{\alpha }$. This latter random variable, as a function of $\alpha$, converges in distribution as $\alpha \downarrow 0$ to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in $\mathbb{D}[0,1]$), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The $\alpha =0$ case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.     

Equivalence classes of regularly varying functions

A useful method to derive limit results for partial maxima and record values of independent, identically distributed random variables is to start from one specific probability distribution and to extend the result for this distribution to a class of distributions.This method involves an extended theory of regularly varying functions. In this paper, equivalence classes of regularly varying functions (in the extended sense) are studied, which is relevant to the problems mentioned above.     

The bivariate normal copula function is regularly varying

We derive the rate of decay of the tail dependence of the bivariate normal distribution and establish its link with regularly varying functions. This result is an initial step in explaining the discrepancy between a zero asymptotic tail dependence coefficient and mass in the tail of a joint distribution.     

Precise large deviations for dependent regularly varying sequences

We study a precise large deviation principle for a stationary regularly varying sequence of random variables. This principle extends the classical results of Nagaev (Theory Probab Appl 14:51–64, 193–208, 1969) and Nagaev (Ann Probab 7:745–789, 1979) for iid regularly varying sequences. The proof uses an idea of Jakubowski (Stoch Proc Appl 44:291–327, 1993; 68:1–20, 1997) in the context of central limit theorems with infinite variance stable limits. We illustrate the principle for stochastic volatility models, real valued functions of a Markov chain satisfying a polynomial drift condition and solutions of linear and non-linear stochastic recurrence equations.     

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.     

Extremal extension of a finitely additive measure

Translated from Matematicheskie Zametki, Vol. 43, No. 1, pp. 25–30, January, 1988.     

Extremal configurations of certain problems on the capacity and harmonic measure

We study certain extremal problems concerning the capacity of a condenser and the harmonic measure of a compact set. In particular, we answer in the negative Tamrazov's question on the minimum of the capacity of a condenser. We find the solution to Dubinin's problem on the maximum of the harmonic measure of a boundary set in the family of domains containing no “long” segments of given inclination. It is also shown that the segment [1-L, 1] has the maximal harmonic measure at the point z=0 among all curves γ={z=z(t), 0≤t≤1}, z(0)=1, that lie in the unit disk and have given length L, 0<L<1. The proofs are based on Baernstein's method of *-functions, Dubinin's dissymmetrization method, and the method of extremal metrics. Bibliography: 21 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 170–195.     

A complete convergence theorem for stationary regularly varying multivariate time series

For a class of stationary regularly varying and weakly dependent multivariate time series (X _n ), we prove the so-called complete convergence result for the space–time point processes of the form \(N_{n} = \sum _{i=1}^{n} \delta _{(i/n, \boldsymbol {X}_{i}/a_{n})}.\) As an application of our main theorem, we give a simple proof of the invariance principle for the corresponding partial maximum process.     

Extremal eigenvalues of measure differential equations with fixed variation

In this paper we will study eigenvalues of measure differential equations which are motivated by physical problems when physical quantities are not absolutely continuous. By taking Neumann eigenvalues of measure differential equations as an example, we will show how the extremal problems can be completely solved by exploiting the continuity results of eigenvalues in weak* topology of measures and the Lagrange multiplier rule for nonsmooth functionals. These results can give another explanation for extremal eigenvalues of Sturm-Liouville operators with integrable potentials.     

Uniform asymptotics for compound Poisson processes with regularly varying jumps and vanishing drift

This paper addresses heavy-tailed large-deviation estimates for the distribution tail of functionals of a class of spectrally one-sided Lévy processes. Our contribution is to show that these estimates remain valid in a near-critical regime. This complements recent similar results that have been obtained for the all-time supremum of such processes. Specifically, we consider local asymptotics of the all-time supremum, the supremum of the process until exiting $[0,\infty )$, the maximum jump until that time, and the time it takes until exiting $[0,\infty )$. The proofs rely, among other things, on properties of scale functions.     

Large deviation principles for random walks with regularly varying distributions of jumps

We establish the local and so-called “extended” large deviation principles (see [1, 2]) for random walks whose jumps fail to satisfy Cramér’s condition but have distributions varying regularly at infinity.     

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.