One- versus multi-component regular variation and extremes of Markov trees

General regular variation, Popa groups and quantifier weakening

We consider an n-player finite strategic game. The payoff vector of each player is a random vector whose distribution is not completely known. We assume that the distribution of a random payoff vector of each player belongs to a distributional uncertainty set. We define a distributionally robust chance-constrained game using worst-case chance constraint. We consider two types of distributional uncertainty sets. We show the existence of a mixed strategy Nash equilibrium of a distributionally robust chance-constrained game corresponding to both types of distributional uncertainty sets. For each case, we show a one-to-one correspondence between a Nash equilibrium of a game and a global maximum of a certain mathematical program.

We study the relations between the tail order of copulas and hidden regular variation (HRV) on subcones generated by order statistics. Multivariate regular variation (MRV) and HRV deal with extremal dependence of random vectors with Pareto-like univariate margins. Alternatively, if one uses a copula to model the dependence structure of a random vector then the upper exponent and tail order functions can be used to capture the extremal dependence structure. After defining upper exponent functions on a series of subcones, we establish the relation between the tail order of a copula and the tail indexes for MRV and HRV. We show that upper exponent functions of a copula and intensity measures of MRV/HRV can be represented by each other, and the upper exponent function on subcones can be expressed by a Pickands-type integral representation. Finally, a mixture model is given with the mixing random vector leading to the finite-directional measure in a product-measure representation of HRV intensity measures.

On asymptotic solutions of Friedmann equations

Let be weak kth records obtained from a sequence of independent variables with a common discrete distribution. In this paper, asymptotic properties of as are discussed by using the theory of regular variation.

Limiting behavior of the ratio of [formula omitted]th records

Let U d = (U 1,…, U d )⊤, d ≥ 2 be a random vector uniformly distributed on the unit sphere of ℝ d , and let A ∈ ℝ d×d be a non-singular matrix. Consider an elliptical random vector X = (X 1,…, X d )⊤ with stochastic representation R A ⊤ U d where the positive random radius R is independent of U d , and let X I = (X i , i ∈ I)⊤, X J = (X i , i ∈ J)⊤ be two vectors with non-empty disjoint index sets I, J, I ∪ J = {1,…, d}. Motivated by the Gaussian approximation of the conditional distribution of bivariate spherical random vectors obtained in Berman[1] we derive in this paper a Gaussian approximation for the conditional distribution X I | X J = u J , u ∈ ℝ d as u J tends to a boundary point provided that the random radius R has distribution function in the Gumbel max-domain of attraction. Further, we generalise Berman's result to the multivariate elliptical setup.

A Pareto distribution has the property that any tail of the distribution has the same shape as the original distribution. The exponential distribution and the uniform distribution have the tail property too. The tail property characterizes the univariate generalized Pareto distributions. There are three classes of univariate GPDs: Pareto distributions, power laws, and the exponential distribution. All these distributions extend to infinite measures. The tail property translates into a group of symmetries for these infinite measures: translations for the exponential law; multiplications for the Pareto and power laws. In the multivariate case, for cylinder symmetric measures in dimension d >_ 3, there are seven classes of measures with the tail property, corresponding to five symmetry groups. The second part of this paper establishes this classification. The first part introduces the probabilistic setting, and discusses the associated geometric theory of multiparameter regular variation. We prove a remarkable result about a class of multiparameter slowly varying functions introduced in Ostrogorski [1995].

Regular variation of distributional tails is known to be preserved by various linear transformations of some random structures. An inverse problem for regular variation aims at understanding whether the regular variation of a transformed random object is caused by regular variation of components of the original random structure. In this paper we build on previous work, and derive results in the multivariate case and in situations where regular variation is not restricted to one particular direction or quadrant.

The posterior distribution of a Dirichlet process from $N$ quantal responses at $r$ dosage levels has been recognized by Antoniak as a mixture of Dirichlet processes. The purpose of this paper is to develop a systematic procedure for computing finite-dimensional distributions of such mixtures which can be equivalently expressed as multivariate beta distributions with random parameter vectors. It is shown that the sequence of random parameter vectors of the updated beta posteriors from observations at increasing dosage levels evolves in a manner which is described by $r$ separate Markov chains. This description is then used to derive the asymptotic posterior distribution. The weak limits of the relevant Markov chains are shown to be solutions of certain stochastic differential equations and the random parameter vector of the posterior beta distribution is shown to be asymptotically normal, the mean vector and covariance matrix of which are given by recursion formulas.

For a random vector $X$ with a fixed distribution $\mu$ we construct a class of distributions ${\cal M}(\mu)= \{ \mu\circ\lambda: \lambda\in{\cal P}\}$, which is the class of all distributions of random vectors $X {\mit\Theta}$, where $ {\mit\Theta}$ is i

Let $(X,Y)$ be a bivariate random vector. The estimation of a probability of the form $P(Y\leq y \mid X >t) $ is challenging when $t$ is large, and a fruitful approach consists in studying, if it exists, the limiting conditional distribution of the random vector $(X,Y)$, suitably normalized, given that $X$ is large. There already exists a wide literature on bivariate models for which this limiting distribution exists. In this paper, a statistical analysis of this problem is done. Estimators of the limiting distribution (which is assumed to exist) and the normalizing functions are provided, as well as an estimator of the conditional quantile function when the conditioning event is extreme. Consistency of the estimators is proved and a functional central limit theorem for the estimator of the limiting distribution is obtained. The small sample behavior of the estimator of the conditional quantile function is illustrated through simulations.

In this article, we study a nonstandard bivariate renewal risk model with dependent claim sizes, where two kinds of claim sizes constitute a sequence of independent and identically distributed random vectors satisfying a certain dependence structure, two claim-number processes are not necessarily independent. When the marginal distributions of the random vectors belong to the intersection of the class of long-tailed distributions and the class of the dominated variation distributions, we obtain joint tail asymptotic formulas of discounted total claims. The result shows that the dependence between two kinds of claims has an effect on the joint asymptotic tail probability of discounted total claims.

Cumulative damage and times of occurrence for a multicomponent system: A discrete time approach

Asymptotic properties of a shot noise sequence, {Xj = ∑ i ≤ jh(τj − τi)Ai}, are considered, where the marginal distribution of the Ai’s is regularly varying at infinity with negative index − α. The tail distribution of the Xj’s is studied with regard to higher order tail area expansions, validity of the Von Mises condition and the inheritability of the second order regular variation property. The Hill’s estimator for the tail index is shown to be asymptotically normally distributed provided the impulse response function h satisfies a mild integrability condition and the distribution of the Xj’s satisfies some regularity conditions.