A conditional limit theorem for a bivariate representation of a univariate random variable and conditional extreme values

Abstract

We first consider a real random variable X represented through a random pair (R, T ) and a deterministic function u as X = R · u(T ). Under quite weak assumptions we prove a limit theorem for (R, T ) given X > x, as x tends to infinity. The novelty of our paper is to show that this theorem for the representation of the univariate random variable X permits us to obtain in an elegant manner conditional limit theorems for random pairs (X, Y ) = R · (u(T ), v(T )) given that X is large. Our approach allows to deduce new results as well as to recover under considerably weaker assumptions results obtained previously in the literature. Consequently, it provides a better understanding and systematization of limit statements for the conditional extreme value models.

The diameter of an elliptical cloud

“…In Section 2, we will define elliptical vectors and state our main results. In section 2.1, extending the results of [BS13], we will show that the realizations of a d-dimensional random elliptical vector with large norm are localized on a subspace of R d whose dimension is the multiplicity of the largest eigenvalue of the covariance matrix. This result will be crucial to prove our main results which are stated in Section 3.…”

“…This model includes the previous one if the function g takes values in the unit sphere. These models were used by [FS10] and [BS13] in the investigation of conditional limit laws of a bivariate vector given that one component is extreme. In such a model, the behavior of the vector given that its norm is large and the behavior of the diameter will be determined by the maxima of the function g .…”

“…The main reason is that if a random variable T is in the domain of attraction of the Gumbel distribution, then for any α > 1, This implies that for T U to be large, U must be very close to its maximum. The full strength of this remark was recently exploited in [BS13] who obtained the rate of convergence of U towards its maximum when the product T U is large and the conditional limiting distribution of the difference between U and its maximum, suitably renormalized. This property explains deeply the conditional limits obtained in [FS10].…”

Explicit results on conditional distributions of generalized exponential mixtures

“…for all 0 < z < ∞ and some positive auxiliary function ψ. More precisely, the result in Corollary 2(i) implies that the auxiliary function can be the constant ψ ≡ 1/λ n ; for a detailed analysis of type-distributions and their relevance in the field of extreme value theory see, e.g., [6,24]. Our results on conditional distributions for sums and subset sums of exponential variables might also be of interest for the analysis of phase-type distributions, as they are important representatives of this class.…”

Extremes of randomly scaled Gumbel risks

“…The study of products of rvs is of interest for numerous applications, see e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13]. We mention below three with Y 1 > 0 being independent of Y 2 .…”