Enkelejd Hashorva

Asymptotics of random contractions

In this paper we discuss the asymptotic behaviour of random contractions X=RS, where R, with distribution function F, is a positive random variable independent of S[set membership, variant](0,1). Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of X assuming that F is in the max-domain of attraction of an extreme value distribution and the distribution function of S satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.

Tail asymptotics under beta random scaling

Abstract Let X , Y , B be three independent random variables such that X has the same distribution function as YB. Assume that B is a beta random variable with positive parameters α , β and Y has distribution function H with H ( 0 ) = 0 . In this paper we derive a recursive formula for calculation of H, if the distribution function H α , β of X is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y, which is closely related to asymptotics of Weyl fractional-order integral operators. We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and H α , β , respectively, and the conditional limiting distribution of bivariate elliptical distributions.

On the number of near-maximum insurance claim under dependence

Abstract Let X1,X2,… be random claim sizes with continuous distribution functions F and N(·) an independent point process on [0,∞). Denote by XN([0,t]):N([0,t]) the maximal claim size occurring during the time interval [0,t],t>0 and K t ( a ) the number of claims that excess the random barrier XN([0,t])−at, with at>0. For iid claim sizes, both distributional and asymptotic properties of K t ( a ) are investigated in Li and Pakes (2001) [Li, Y., Pakes, A.G., Insur.: Math. Econ. 28 (3), 309–323]. In this paper, by dropping the iid assumption we extend and simplify limit results of Li and Pakes (2001). Further, we show that K t ( a ) / t is a strongly consistent estimator of certain tail probability.

Exact tail asymptotics in bivariate scale mixture models

Let (X, Y ) = (RU1, RU2) be a given bivariate scale mixture random vector, with R > 0 being independent of the bivariate random vector (U1, U2). In this paper we derive exact asymptotic expansions of the joint survivor probability of (X, Y ) assuming that R has distribution function in the Gumbel max-domain of attraction and (U1, U2) has a specific tail behaviour around some absorbing point. We apply our results to investigate the asymptotic behaviour of joint conditional excess distribution and the asymptotic independence for two models of bivariate scale mixture distributions. Furthermore for our models we derive an expression of the residual dependence index η.

On multivariate Gaussian tails

Let {Xn, n≥1} be a sequence of standard Gaussian random vectors in ℝd,d ≥ 2. In this paper we derive lower and upper bounds for the tail probabilityP{Xn>tn} withtn ∈ ℝd some threshold. We improve for instance bounds on Mills ratio obtained by Savage (1962,J. Res. Nat. Bur. Standards Sect. B,66, 93–96). Furthermore, we prove exact asymptotics under fairly general conditions on bothXn andtn, as ‖tn‖→∞ where the correlation matrix Σn ofXn may also depend onn.

On the supremum of γ-reflected processes with fractional Brownian motion as input

Abstract Let { X H ( t ) , t ≥ 0 } be a fractional Brownian motion with Hurst index H ∈ ( 0 , 1 ] and define a γ -reflected process W γ ( t ) = X H ( t ) − c t − γ inf s ∈ [ 0 , t ] ( X H ( s ) − c s ) , t ≥ 0 with c > 0 , γ ∈ [ 0 , 1 ] two given constants. In this paper we establish the exact tail asymptotic behaviour of M γ ( T ) = sup t ∈ [ 0 , T ] W γ ( t ) for any T ∈ ( 0 , ∞ ] . Furthermore, we derive the exact tail asymptotic behaviour of the supremum of certain non-homogeneous mean-zero Gaussian random fields.

Gaussian Approximation of Conditional Elliptical Random Vectors

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 Bermans result to the multivariate elliptical setup.

On the residual dependence index of elliptical distributions

The residual dependence index of bivariate Gaussian distributions is determined by the correlation coefficient. This tail index is of certain statistical importance when extremes and related rare events of bivariate samples with asymptotic independent components are being modeled. In this paper we calculate the partial residual dependence indices of a multivariate elliptical random vector assuming that the associated random radius has distribution function in the Gumbel max-domain of attraction. Furthermore, we discuss the estimation of these indices when the associated random radius possesses a Weibull-tail distribution.

Asymptotics of a boundary crossing probability of a Brownian bridge with general trend

Let us consider a signal-plus-noise model γh(z)+B0(z), z ∈ [0,1], where γ > 0, h: [0,1] → ℝ, and B0 is a Brownian bridge. We establish the asymptotics for the boundary crossing probability of the weighted signal-plus-noise model for γ→∞, that is P (supzε [0,1]w(z)(γ h(z)+B0(z))>c), for γ→∞, (1) where w: [0,1]→ [0,∞ is a weight function and c>0 is arbitrary. By the large deviation principle one gets a result with a constant which is the solution of a minimizing problem. In this paper we show an asymptotic expansion that is stronger than large deviation. As byproduct of our result we obtain the solution of the minimizing problem occurring in the large deviation expression. It is worth mentioning that the probability considered in (1) appears as power of the weighted Kolmogorov test applied to the test problem H0: h≡ 0 against the alternative K: h>0 in the signal-plus-noise model.

Exact asymptotics for Boundary crossings of the brownian bridge with trend with application to the Kolmogorov test

We consider a boundary crossing probability of a Brownian bridgeB0 and a piecewise linear boundary functionu(t)−γh(t). The main result of this paper is an asymptotic expansion for γ→∞ of the boundary crossing probability thatB0(t) is larger than the piecewise linear boundary functionu(t)−γh(t) for somet. Such probabilities occur for instance in the context of change point problems when the Kolmogorov test is used. Examples are discussed showing that the approximation is rather accurate even for small positive γ values.