Estimation of conditional laws given an extreme component

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.

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.

Let \((U_{ni},V_{ni}), 1 \le i \le n, n\ge 1\) be a triangular array of independent bivariate elliptical random vectors with the same distribution function as \(\bigl( S_1, \rho_n S_1 + \sqrt{1- \rho_n^2}S_2\bigr), \rho_n \in (0,1)\) where \((S_1, S_2)\) is a bivariate spherical random vector. Under assumptions on the speed of convergence of \(\rho_n\to 1\) we show in this paper that the maxima of this triangular array is in the max-domain of attraction of a new max-id. distribution function \(H_{\alpha,\lambda}\), provided that \(\sqrt{S_1^2+S_2^2}\) has distribution function in the max-domain of attraction of the Weibull distribution function \(\Psi_\alpha\).

A Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called a tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one-component regular variation. Changing the location of the root, that is, changing the conditioning variable, yields a different tail tree. When the tails of the marginal distributions of the conditioning variables are balanced, these tail trees are connected by a formula that generalizes the time change formula for regularly varying stationary time series. The formula is most easily understood when the various one-component regular variation statements are tied up into a single multi-component statement. The theory of multi-component regular variation is worked out for general random vectors, not necessarily Markov trees, with an eye towards other models, graphical or otherwise.

. Kullback and Leibler (1951) introduced Kullback–Leibler divergence to measure the disparity between two random variables. In this article, we introduce a bivariate extension of Kullback–Leibler divergence and study its various properties. Bivariate Kullback–Leibler divergence is a useful tool to compute the disparity between two bivariate random vectors. Some useful bounds are obtained for bivariate Kullback–Leibler divergence, and some are based on entropy and inaccuracy measures. A non parametric estimator for the measure is also proposed, and simulation studies are carried out for validation. Finally, the proposed measure is applied to a real-life data in survival studies.

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

Continuation refers to the operation by which the cumulative distribution\nfunction of a discontinuous random vector is made continuous through\nmultilinear interpolation. The copula that results from the application of this\ntechnique to the classical empirical copula is either called the multilinear or\nthe checkerboard copula. As shown by Genest and Ne\\v{s}lehov\\'{a} (Astin Bull.\n37 (2007) 475-515) and Ne\\v{s}lehov\\'{a} (J. Multivariate Anal. 98 (2007)\n544-567), this copula plays a central role in characterizing dependence\nconcepts in discrete random vectors. In this paper, the authors establish the\nasymptotic behavior of the empirical process associated with the multilinear\ncopula based on $d$-variate count data. This empirical process does not\ngenerally converge in law on the space $\\mathcal {C}([0,1]^d)$ of continuous\nfunctions on $[0,1]^d$, equipped with the uniform norm. However, the authors\nshow that the process converges in $\\mathcal{C}(K)$ for any compact\n$K\\subset\\mathcal{O}$, where $\\mathcal{O}$ is a dense open subset of $[0,1]^d$,\nwhose complement is the Cartesian product of the ranges of the marginal\ndistribution functions. This result is sufficient to deduce the weak limit of\nmany functionals of the process, including classical statistics for monotone\ntrend. It also leads to a powerful and consistent test of independence which is\napplicable even to sparse contingency tables whose dimension is sample size\ndependent.\n

Convergence of averages of scaled functions of I(1) linear processes

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.

The need for calculating and characterizing singular normal distributions arises in a natural way when considering chance constraints of the type P(Az <= b(x) >= p, where A is a rectangular matrix having more rows than columns, b is some function and z is a random vector having some nondegenerate multivariate normal distribution. Such situation is typical, for instance, in stochastic networks, where a comparatively small random vector may induce a possibly large number of linear inequality constraints. Passing to the transformed random variable q:=Az, the constraint can be equivalently rewritten as F(b(x))>= p, where F is the distribution function of q. In contrast to the original random vector z, the transformed vector q has a singular normal distribution. The talk demonstrates how to get back from here to (a sum of) regular normal distributions under a full rank regularity condition. This allows for an efficient calculation of singular normal distributions and provides a numerical method which outperforms competing procedures in moderate dimensions. Computational results for test examples are provided for the sake of comparison. In general, if the mentioned regularity condition is violated, then the singular normal distribution function F may even lack continuity. The talk provides an equivalent criterion for Lipschitz continuity of F and characterizes differentiability and subdifferentiability of F.

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities $\Pr(n^{-1/2}\sum_{i=1}^n X_i\in A)$ where $X_1,\dots,X_n$ are independent random vectors in $\mathbb{R}^p$ and $A$ is a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if $p=p_n\to \infty$ as $n \to \infty$ and $p \gg n$; in particular, $p$ can be as large as $O(e^{Cn^c})$ for some constants $c,C>0$. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of $X_i$. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.

In this paper, we consider functional limit theorems for Poisson cluster processes. We first present a maximal inequality for Poisson cluster processes. Then we establish a functional central limit theorem under the second moment and a functional moderate deviation principle under the Cramér condition for Poisson cluster processes. We apply these results to obtain a functional moderate deviation principle for linear Hawkes processes.

We study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.

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.