Tail Asymptotics of Deflated Risks

Random deflated risk models have been considered in recent literatures. In this paper, we investigate second-order tail behavior of the deflated risk X=RS under the assumptions of second-order regular variation on the survival functions of the risk R and the deflator S. Our findings are applied to approximation of Value at Risk, estimation of small tail probability under random deflation and tail asymptotics of aggregated deflated riskComment: 2

Representations of max-stable processes via exponential tilting

The recent contribution Dieker & Mikosch (2015) [1] obtained important representations of max-stable stationary Brown-Resnick random fields ζZ with a spectral representation determined by a Gaussian process Z. With motivations from \cite{DM} we derive for some general Z, representations for ζZ via exponential tilting of Z. Our main findings concern a) Dieker-Mikosch representations of max-stable processes, b) two-sided extensions of stationary max-stable processes, c) inf-argmax representation of any max-stable distribution, and d) new formulas for generalised Pickands constants. Our applications include new conditions for the stationarity of ζZ, a characterisation of Gaussian random vectors and an alternative proof of Kabluchko's characterisation of Gaussian processes with stationary increments

Estimation of conditional laws given an extreme component

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.Comment: 32 pages, 5 figur

Maxima and minima of complete and incomplete stationary sequences

In the seminal contribution [R. A. Davis, Maxima and minima of stationary sequences, Ann. Probab. 7(3) (1979), pp. 453-460.] the joint weak convergence of maxima and minima of weakly dependent stationary sequences is derived under some mild asymptotic conditions. In this paper we address additionally the case of incomplete samples assuming that the average proportion of incompleteness converges in probability to some random variable P. We show the joint weak convergence of the maxima and the minima of both complete and incomplete samples. It turns out that the maxima and the minima are asymptotically independent when P is a deterministic constant

On Piterbarg Max-Discretisation Theorem for Standardised Maximum of Stationary Gaussian Processes

With motivation from Husler (Extremes 7:179-190, 2004) and Piterbarg (Extremes 7:161-177, 2004) in this paper we derive the joint limiting distribution of standardised maximum of a continuous, stationary Gaussian process and the standardised maximum of this process sampled at discrete time points. We prove that these two random sequences are asymptotically complete dependent if the grid of the discrete time points is sufficiently dense, and asymptotically independent if the grid is sufficiently sparse. We show that our results are relevant for computational problems related to discrete time approximation of the continuous time maximum

Extremes and First Passage Times of Correlated Fractional Brownian Motions

Let {X-i (t), t >= 0}, i = 1, 2 be two standard fractional Brownian motions being jointly Gaussian with constant cross-correlation. In this paper, we derive the exact asymptotics of the joint survival function P {sups(is an element of)[(0,1]) X-1(s) > u, sup(t is an element of)[(0,1]) X-2(t) > u} as u ->infinity. A novel finding of this contribution is the exponential approximation of the joint conditional first passage times of X-1, X-2. As a by-product, we obtain generalizations of the Borell-TIS inequality and the Piterbarg inequality for 2-dimensional Gaussian random fields. Keywords Borell-TIS inequality; Extremes; First passage times; Fractional Brownian motion; Gaussian random fields; Piterbarg inequality