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Asymptotics of the convex hull of spherical samplesDec 25 2014In this paper we consider the convex hull of a spherically symmetric sample in $R^d$. Our main contributions are some new asymptotic results for the expectation of the number of vertices, number of facets, area and the volume of the convex hull assuming ... More

Boundary non-crossings of Brownian pillowSep 26 2008Let B_0(s,t) be a Brownian pillow with continuous sample paths, and let h,u:[0,1]^2\to R be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability \psi(u;h):=P{B_0(s,t)+h(s,t) \le u(s,t), \forall ... More

Extremes of Aggregated Dirichlet RisksApr 12 2014Dec 11 2014The class of Dirichlet random vectors is central in numerous probabilistic and statistical applications. The main result of this paper derives the exact tail asymptotics of the aggregated risk of powers of Dirichlet random vectors when the radial component ... More

Representations of Max-Stable Processes via Exponential TiltingMay 10 2016Jul 02 2016The recent contribution Dieker & Mikosch (2015) [1] derived important representations of max-stable Brown-Resnick random fields $\zeta_Z$ with a spectral representation determined by a Gaussian random field $Z$. With motivations from [1] we derive for ... More

Minima and maxima of elliptical arrays and spherical processesJul 23 2013In this paper, we investigate first the asymptotics of the minima of elliptical triangular arrays. Motivated by the findings of Kabluchko (Extremes 14 (2011) 285-310), we discuss further the asymptotic behaviour of the maxima of elliptical triangular ... More

Exact Tail Asymptotics of Dirichlet DistributionsApr 01 2009Apr 19 2010Let X be a generalised symmetrised Dirichlet random vector in R^k, and let u_n be thresholds such that P{X> u_n} tends to 0 as n goes infinity. In this paper we derive an exact asymptotic expansion of P{X> u_n} assuming that the associated random radius ... More

Conditional Limits of W_p scale Mixture DistributionsOct 31 2008Mar 24 2009In this paper we introduce the class of W_p scale mixture random vectors with a particular radial decomposition and a independent splitting property specified by some random variable W_p, and a positive constant p. We derive several conditional limit ... More

Asymptotics for Kotz Type III Elliptical DistributionsNov 05 2008In this paper we derive the tail asymptotics of a Kotz Type III elliptical random vector. As an application of our asymptotic expansion we derive an approximation for the conditional excess distribution. Furthermore, we discuss the asymptotic dependence ... More

Conditional Limit Results for Type I Polar DistributionsOct 08 2008Let (S_1,S_2)=(R \cos(\Theta), R \sin (\Theta)) be a bivariate random vector with associated random radius R which has distribution function $F$ being further independent of the random angle \Theta. In this paper we investigate the asymptotic behaviour ... More

Exact Asymptotics of Bivariate Scale Mixture DistributionsApr 06 2009Apr 19 2010Let (RU_1, R U_2) be a given bivariate scale mixture random vector, with R>0 being independent of the bivariate random vector (U_1,U_2). In this paper we derive exact asymptotic expansions of the tail probability P{RU_1> x, RU_2> ax}, a \in (0,1] as x ... More

On the asymptotic distribution of certain bivariate reinsurance treatiesMar 30 2006May 17 2006Let (X_n,Y_n), n\ge 1 be bivariate random claim sizes with common distribution function F and let N(t), t \ge 0 be a stochastic process which counts the number of claims that occur in the time interval [0,t], t\ge 0. In this paper we derive the joint ... More

Tail Behaviour of Weighted Sums of Order Statistics of Dependent RisksAug 06 2014Let $X_{1},\ldots ,X_{n}$ be $n$ real-valued dependent random variables. With motivation from Mitra and Resnick (2009), we derive the tail asymptotic expansion for the weighted sum of order statistics $X_{1:n}\leq \cdots \leq X_{n:n}$ of $X_{1},\ldots ... More

Limit Laws for Maxima of Contracted Stationary Gaussian SequencesDec 07 2013The principal results of this contribution are the weak and strong limits of maxima of contracted stationary Gaussian random sequences. Due to the random contraction we introduce a modified Berman condition which is sufficient for the weak convergence ... More

Random Shifting and Scaling of Insurance RisksApr 12 2014Random shifting typically appears in credibility models whereas random scaling is often encountered in stochastic models for claim sizes reflecting the time-value property of money. In this article we discuss some aspects of random shifting and random ... More

Extremes of alpha(t)-locally Stationary Gaussian Random FieldsSep 01 2013This contribution derives the exact asymptotic behaviour of the supremum of alpha(t)-locally stationary Gaussian random fields over a finite hypercube. We present two applications of our result; the first one deals with extremes of ggregate multifractional ... More

Berman's inequality under random scalingSep 24 2013Apr 23 2014Berman's inequality is the key for establishing asymptotic properties of maxima of Gaussian random sequences and supremum of Gaussian random fields. This contribution shows that, asymptotically an extended version of Berman's inequality can be established ... More

Boundary Non-Crossings of Additive Wiener FieldsFeb 11 2014Jun 25 2014Let $W_i=\{W_i(t), t\in \mathbb{R}_+\}, i=1,2$ be two Wiener processes and $W_3=\{W_3(\mathbf{t}), \mathbf{t}\in \mathbb{R}_+^2\}$ be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for ... More

Asymptotics for a discrete-time risk model with the emphasis on financial riskApr 23 2014This paper focuses on a discrete-time risk model in which both insurance risk and financial risk are taken into account. We study the asymptotic behaviour of the ruin probability and the tail probability of the aggregate risk amount. Precise asymptotic ... More

Tail approximation for reinsurance portfolios of Gaussian-like risksMay 03 2014We consider two different portfolios of proportional reinsurance of the same pool of risks. This contribution is concerned with Gaussian-like risks, which means that for large values the survival function of such risks is, up to a multiplier, the same ... More

Joint Limiting Distribution of Minima and Maxima of Complete and Incomplete Samples of Stationary SequencesSep 24 2013In the seminal contribution [4] 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 ... More

Approximation of a random process with variable smoothnessJun 06 2012We consider the rate of piecewise constant approximation to a locally stationary process $X(t),t\in [0,1]$, having a variable smoothness index $\alpha(t)$. Assuming that $\alpha(\cdot)$ attains its unique minimum at zero and satisfies the regularity condition, ... More

Extremes of homogeneous Gaussian random fieldsDec 10 2013Let $\{X(s,t):s,t\geqslant 0\}$ be a centered homogeneous Gaussian field with a.s. continuous sample paths and correlation function $r(s,t)=Cov(X(s,t),X(0,0))$ such that \[r(s,t)=1-|s|^{\alpha_1}-|t|^{\alpha_2}+o(|s|^{\alpha_1}+|t|^{\alpha_2}), \quad ... More

Multivariate Extremes of a Random Number of MaximaDec 27 2017Aug 07 2018The classical multivariate extreme-value theory concerns the modeling of extremes in a multivariate random sample. The observations with large values in at least one component is an example of this. The theory suggests the use of max-stable distributions. ... More

On the gamma-reflected processes with fBm inputFeb 11 2014Define a $\gamma$-reflected process $W_\gamma(t)=Y_H(t)-\gamma\inf_{s\in[0,t]}Y_H(s)$, $t\ge0$ with input process $\{Y_H(t), t\ge 0\}$ which is a fractional Brownian motion with Hurst index $H\in (0,1)$ and a negative linear trend. In risk theory $R_\gamma(t)=u-W_\gamma(t), ... More

Tail measure and tail spectral process of regularly varying time seriesOct 23 2017Jul 14 2018The goal of this paper is an exhaustive investigation of the link between the tail measure of a regularly varying time series and its spectral tail process, independently introduced in Owada and Samorodnitsky (2012) and Basrak and Segers (2009). Our main ... More

Ruin probabilities and passage times of $γ$-reflected Gaussian processes with stationary incrementsNov 30 2015For a given centered Gaussian process with stationary increments $\{X(t), t\geq 0\}$ and $c>0$, let $$ W_\gamma(t)=X(t)-ct-\gamma\inf_{0\leq s\leq t}\left(X(s)-cs\right), \quad t\geq 0$$ denote the $\gamma$-reflected process, where $\gamma\in (0,1)$. ... More

Extremes of a class of nonhomogeneous Gaussian random fieldsMay 12 2014Mar 15 2016This contribution establishes exact tail asymptotics of $\sup_{(s,t)\in\mathbf{E}}$ $X(s,t)$ for a large class of nonhomogeneous Gaussian random fields $X$ on a bounded convex set $\mathbf{E}\subset\mathbb{R}^2$, with variance function that attains its ... More

Tail asymptotics of randomly weighted large risksMay 03 2014Jun 23 2014In this paper we are concerned with a sample of asymptotically independent risks. Tail asymptotic probabilities for linear combinations of randomly weighted order statistics are approximated under various assumptions, where the individual tail behaviour ... More

Random Scaling of Gumbel RisksDec 26 2013Jun 23 2014In this paper we consider the product of two positive independent risks $Y_1$ and $Y_2$. If $Y_1$ is bounded and $Y_2$ has distribution in the Gumbel max-domain of attraction with some auxiliary function which is regularly varying at infinity, then we ... More

On the Supremum of gamma-reflected Processes with Fractional Brownian Motion as InputJun 09 2013Let $X_H(t), t\ge 0$ be a fractional Brownian motion with Hurst index $H\in(0,1}$ and define a gamma-reflected process $W_\Ga(t)=X_H(t)-ct-\gammainf_{s\in[0,t]}\left(X_H(s)-cs \right)$, $t\ge0$ with $c>0,\gamma \in [0,1]$ two given constants. In this ... More

Simultaneous Ruin Probability for Two-Dimensional Brownian and Lévy Risk ModelsNov 11 2018The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite-time horizon. This is not the case for the simultaneous ruin probability in two-dimensional Brownian risk model. Resorting on asymptotic ... More

Uniform tail approximation of homogenous functionals of Gaussian fieldsJul 05 2016Jun 08 2017Let $X(t),t\in R^d$ be a centered Gaussian random field with continuous trajectories and set $\xi_u(t)= X(f(u)t),t\in R^d$ with $f$ some positive function. Classical results establish the tail asymptotics of $P\{ \Gamma(\xi_u) > u\}$ as $u\to \infty$ ... More

Gaussian Approximation of Perturbed Chi-Square RisksSep 19 2013In this paper we show that the conditional distribution of perturbed chi-quare risks can be approximated by certain distributions including the Gaussian ones. Our results are of interest for conditional extreme value models and multivariate extremes as ... More

Parisian Ruin of Self-similar Gaussian Risk ProcessesMay 12 2014In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian ... More

Maxima of a triangular array of multivariate Gaussian sequenceFeb 23 2014It is known that the normalized maxima of a sequence of independent and identically distributed bivariate normal random vectors with correlation coefficient $\rho \in (-1,1)$ is asymptotically independent, which may seriously underestimate extreme probabilities ... More

Bounds and large deviation results for boundary non-crossing probabilities of Gaussian processesMar 14 2019We study boundary non-crossing probabilities $$ P_{f,u} := \mathrm{P}\big(\forall t\in \mathbb T\ X_t + f(t)\le u(t)\big) $$ for continuous centered Gaussian process $X$ indexed by arbitrary compact separable metrizable space $\mathbb T$. We obtain upper ... More

Asymptotics of the Norm of Elliptical Random VectorsDec 22 2008Oct 13 2009In this paper we consider elliptical random vectors X in R^d,d>1 with stochastic representation A R U where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A is a given matrix. ... More

On the Probability of Conjunctions of Stationary Gaussian ProcessesDec 26 2013Jan 18 2014Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent centered stationary Gaussian processes with unit variance and almost surely continuous sample paths. For given positive constants $u,T$, define the set of conjunctions $C_{[0,T],u}:=\{t\in [0,T]: \min_{1 ... More

Representations of Max-Stable Processes via Exponential TiltingMay 10 2016Jun 10 2017The recent contribution Dieker & Mikosch (2015) [1] obtained important representations of max-stable stationary Brown-Resnick random fields $\zeta_Z$ with a spectral representation determined by a Gaussian process $Z$. With motivations from \cite{DM} ... More

On bivariate lifetime modelling in life insurance applicationsJan 17 2016Insurance and annuity products covering several lives require the modelling of the joint distribution of future lifetimes. In the interest of simplifying calculations, it is common in practice to assume that the future lifetimes among a group of people ... More

On Beta-Product ConvolutionsJan 26 2010Aug 16 2010Let R be a positive random variable independent of S which is beta distributed. In this paper we are interested on the relation between the distribution function of R and that of RS. For this model we derive first some distributional properties, and then ... More

On the residual dependence index of elliptical distributionsNov 21 2008Aug 18 2009The 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 ... More

Tail Asymptotics and Estimation for Elliptical DistributionsAug 14 2007May 15 2008Let (X,Y) be a bivariate elliptical random vector with associated random radius in the Gumbel max-domain of attraction. In this paper we obtain a second order asymptotic expansion of the joint survival probability P(X > x, Y> y) for x,y large. Further, ... More

Approximation of Sojourn Times of Gaussian ProcessesDec 13 2017We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes $X$, in both continuous- and discrete-time framework. All results obtained here are new for the discrete-time case. In the continuous-time ... More

Extremes of vector-valued Gaussian processes: exact asymptoticsMay 24 2015Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of $$ P\left(\exists_{t \in [0,T]} \forall_{i=1 ... n} X_i(t)> u \right) $$ as $u\to\infty$, ... More

Extremes of Order Statistics of Stationary ProcessesMar 28 2014Aug 06 2014Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a stationary process $\{X(t), t\ge0\}$. For given positive constants $u,T$, define the set of $r$th conjunctions $ C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}$ with $X_{r:n}(t)$ the $r$th largest ... More

Distribution and asymptotics under beta random scalingDec 04 2008May 13 2009Let X,Y,B be three independent random variables such that $X$ has the same distribution function as Y B. Assume that B is a Beta random variable with positive parameters a,b and Y has distribution function H. Pakes and Navarro (2007) show under some mild ... More

On Extremal Index of max-stable stationary processesApr 05 2017In this contribution we discuss the relation between Pickands-type constants defined for certain Brown-Resnick stationary process $W(t),t\in R$ as $$\mathcal{H}_W^\delta= \lim_{T\to\infty} T^{-1} E{ \left(\sup_{t\in \delta Z \cap [0,T]} e^{W(t)}\right) ... More

Piterbarg Theorems for Chi-processes with TrendSep 01 2013Let $\chi_n(t) = (\sum_{i=1}^n X_i^2(t))^{1/2},t\ge0$ be a chi-process with $n$ degrees of freedom where $X_i$'s are independent copies of some generic centered Gaussian process $X$. This paper derives the exact asymptotic behavior of P{\sup_{t\in[0,T]} ... More

Tail Asymptotics of Random Sum and Maximum of Log-Normal RisksJan 18 2014In this paper we derive the asymptotic behaviour of the survival function of both random sum and random maximum of log-normal risks. As for the case of finite sum and maximum investigated in Asmussen and Rojas-Nandaypa (2008) also for the more general ... More

Extremes and first passage times of correlated fBm'sSep 19 2013Feb 25 2014Let $\{X_i(t),t\ge0\}, i=1,2$ 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 $$ \mathbb{P}\{\sup_{s\in[0,1]}X_1(s)>u,\ \sup_{t\in[0,1]}X_2(t)>u\} ... More

Maxima of Skew Elliptical Triangular ArraysDec 07 2013In this paper we investigate the asymptotic behaviour of the componentwise maxima for two bivariate skew elliptical triangular arrays with components given in terms of skew transformations of bivariate spherical random vectors. We find the weak limit ... More

Large Deviations of Shepp Statistics for Fractional Brownian MotionJun 09 2013Define the incremental fractional Brownian field $B_{H}(s+\tau)-B_{H}(s), H\in (0,1)$, where $B_{H}(s)$ is a standard fractional Brownian motion with Hurst index $H\in(0,1)$. In this paper we derive the exact asymptotic behaviour of the maximum $\max_{(\tau,s)\in[0,1]\times[0,T]} ... More

Approximation of passage times of gamma-reflected processes with fBm inputOct 11 2013Define a gamma-reflected process W_\gamma(t)=Y_H(t)-\gamma\inf_{s\in[0,t]}Y_H(s), t\ge0 with input process {Y_H(t), t\ge 0} which is a fractional Brownian motion with Hurst index H\in (0,1) and a negative linear trend. In risk theory {u-W_\gamma(t), t\ge ... More

Uniform tail approximation of homogenous functionals of Gaussian fieldsJul 05 2016Let $X(t),t\in R^d$ be a centered Gaussian random field with continuous trajectories and set $\xi_u(t)= X(f(u)t),t\in R^d$ with $f$ some positive function. Classical results establish the tail asymptotics of $P\{ \Gamma(\xi_u) > u\}$ as $u\to \infty$ ... More

Asymptotics of Random ContractionsJul 31 2010In 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 $S\in (0,1)$. Random contractions appear naturally in insurance and finance. Our principal ... More

Extremes of $γ$-reflected Gaussian process with stationary incrementsNov 30 2015Nov 07 2017For a given centered Gaussian process with stationary increments $\{X(t), t\geq 0\}$ and $c>0$, let $$ W_\gamma(t)=X(t)-ct-\gamma\inf_{0\leq s\leq t}\left(X(s)-cs\right), \quad t\geq 0$$ denote the $\gamma$-reflected process, where $\gamma\in (0,1)$. ... More

Insurance Applications of Some New Dependence Models derived from Multivariate Collective ModelsMar 06 2016Oct 06 2016Consider two different portfolios which have claims triggered by the same events. Their corresponding collective model over a fixed time period is given in terms of individual claim sizes $(X_i,Y_i), i\ge 1$ and a claim counting random variable $N$. In ... More

Gaussian risk models with financial constraintsSep 29 2013In this paper we investigate Gaussian risk models which include financial elements such as inflation and interest rates. For some general models for inflation and interest rates, we obtain an asymptotic expansion of the finite-time ruin probability for ... More

Boundary non-crossing probabilities for fractional Brownian motion with trendSep 29 2013In this paper we investigate the boundary non-crossing probabilities of a fractional Brownian motion considering some general deterministic trend function. We derive bounds for non-crossing probabilities and discuss the case of a large trend function. ... More

Extremal behavior of squared Bessel processes attracted by the Brown-Resnick processNov 14 2013The convergence of properly time-scaled and normalized maxima of independent standard Brownian motions to the Brown-Resnick process is well-known in the literature. In this paper, we study the extremal functional behavior of non-Gaussian processes, namely ... More

Extremes of Gaussian Random Fields with regularly varying dependence structureMay 28 2016Let $X(t), t\in \mathcal{T}$ be a centered Gaussian random field with variance function $\sigma^2(\cdot)$ that attains its maximum at the unique point $t_0\in \mathcal{T}$, and let $M(\mathcal{T}):=\sup_{t\in \mathcal{T}} X(t)$. For $\mathcal{T}$ a compact ... More

Tail Asymptotics of Supremum of Certain Gaussian Processes over Threshold Dependent Random IntervalsNov 22 2013Let $\{X(t),t\ge0\}$ be a centered Gaussian process and let $\gamma$ be a non-negative constant. In this paper we study the asymptotics of $P\{\underset{t\in [0,\mathcal{T}/u^\gamma]}\sup X(t)>u\}$ as $u\to\infty$, with $\mathcal{T}$ an independent of ... More

Asymptotic Expansion of Gaussian Chaos via Probabilistic ApproachJul 22 2013Feb 17 2015For a centered $d$-dimensional Gaussian random vector $\xi =(\xi_1,\ldots,\xi_d)$ and a homogeneous function $h:R^d\to R$ we derive asymptotic expansions for the tail of the Gaussian chaos $h(\xi)$ given the function $h$ is sufficiently smooth. Three ... More

Limit properties of exceedances point processes of scaled stationary Gaussian sequencesOct 19 2013We derive the limiting distributions of exceedances point processes of randomly scaled weakly dependent stationary Gaussian sequences under some mild asymptotic conditions. In the literature analogous results are available only for contracted stationary ... More

Generalized Pickands constants and stationary max-stable processesFeb 04 2016Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, ... More

On Parisian ruin over a finite-time horizonApr 27 2015For a risk process $R_u(t)=u+ct-X(t), t\ge 0$, where $u\ge 0$ is the initial capital, $c>0$ is the premium rate and $X(t),t\ge 0$ is an aggregate claim process, we investigate the probability of the Parisian ruin \[ \mathcal{P}_S(u,T_u)=\mathbb{P}\{\inf_{t\in[0,S]} ... More

Extremes of multidimensional Brownian motion with driftOct 28 2016This paper studies an exact asymptotics of \[ P\{\exists_{t\ge 0} {X}(t)- {\mu}t\in {\mathcal{U}}_u \}, \ \ {\rm as}\ \ u\to\infty, \] where ${X}(t)=(X_1(t),\ldots,X_d(t))^\top,t\ge0$ is a correlated $d$-dimensional Brownian motion, $\mu=(\mu_1,...,\mu_d)^\top ... More

Finite-time ruin probability of aggregate Gaussian processesApr 23 2014Let $\left\{\sum_{i=1}^n \lambda_i X_i(t), t\in [0,T]\right\}$ be an aggregate Gaussian risk process with $X_i, i\leq n$ independent Gaussian processes satisfying Piterbarg conditions and $\lambda_i$'s given positive weights. In this paper we derive exact ... More

Extremal behaviour of hitting a cone by correlated Brownian motion with driftOct 28 2016Jul 10 2017This paper derives an exact asymptotic expression for \[ \mathbb{P}_{\mathbf{x}_u}\{\exists_{t\ge0} \mathbf{X}(t)- \boldsymbol{\mu}t\in \mathcal{U} \}, \ \ {\rm as}\ \ u\to\infty, \] where $\mathbf{X}(t)=(X_1(t),\ldots,X_d(t))^\top,t\ge0$ is a correlated ... More

Limit Laws for Extremes of Dependent Stationary Gaussian ArraysMay 11 2014May 23 2014In this paper we show that the componentwise maxima ofweakly dependent bivariate stationary Gaussian triangular arrays converge in distribution after normalisation to H\"usler-Reiss distribution. Under a strong dependence assumption, we prove that the ... More

On Piterbarg Max-discretisation Theorem for Multivariate Stationary Gaussian ProcessesMay 10 2014Let $\{X(t), t\geq0\}$ be a stationary Gaussian process with zero-mean and unit variance. A deep result derived in Piterbarg (2004), which we refer to as Piterbarg's max-discretisation theorem gives the joint asymptotic behaviour ($T\to \infty$) of the ... More

Second order asymptotics of aggregated log-elliptical riskMay 03 2014In this paper we establish the error rate of first order asymptotic approximation for the tail probability of sums of log-elliptical risks. Our approach is motivated by extreme value theory which allows us to impose only some weak asymptotic conditions ... More

Tail asymptotic of Weibull-type risksMay 10 2014In this paper we derive the tail asymptotics of the product of two dependent Weibull-type risks, which is of interest in various statistical and applied probability problems. Our results extend some recent findings of Schlueter and Fischer (2012) and ... More

Efficient simulation of tail probabilities for sums of log-elliptical risksMay 03 2014In the framework of dependent risks it is a crucial task for risk management purposes to quantify the probability that the aggregated risk exceeds some large value u. Motivated by Asmussen et al. (2011) in this paper we introduce a modified Asmussen-Kroese ... More

Higher-order expansions of distributions of maxima in a Hüsler-Reiss modelFeb 23 2014The max-stable H\"usler-Reiss distribution which arises as the limit distribution of maxima of bivariate Gaussian triangular arrays has been shown to be useful in various extreme value models. For such triangular arrays, this paper establishes higher-order ... More

Tail Asymptotics of Deflated RisksMay 12 2013Random 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 ... More

On maximum of Gaussian process with unique maximum point of its varianceJan 28 2019Gaussian random processes which variances reach theirs maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximums of theirs trajectories have been evaluated using Double Sum Method under the ... More

Some Mathematical Aspects of Price OptimisationMay 19 2016Calculation of an optimal tariff is a principal challenge for pricing actuaries. In this contribution we are concerned with the renewal insurance business discussing various mathematical aspects of calculation of an optimal renewal tariff. Our motivation ... More

Piterbarg's max-discretisation theorem for stationary vector Gaussian processes observed on different gridsOct 07 2014In this paper we derive Piterbarg's max-discretisation theorem for two different grids considering centered stationary vector Gaussian processes. So far in the literature results in this direction have been derived for the joint distribution of the maximum ... More

Tail Asymptotic Expansions for L-StatisticsFeb 25 2014In this paper, we derive higher-order expansions of $L$-statistics of independent risks $X_1, \ldots, X_n$ under conditions on the underlying distribution function $F$. The new results are applied to derive the asymptotic expansions of ratios of two kinds ... More

Asymptotics of maxima of strongly dependent Gaussian processesApr 23 2014Let $\{X_{n}(t), t\in[0,\infty)\}, n\in\mathbb{N}$ be a sequence of centered dependent stationary Gaussian processes. The limit distribution of $\sup_{t\in[0,T(n)]}|X_{n}(t)|$ is established as $r_{n}(t)$, the correlation function of $X_{n}$ satisfies ... More

Extremes and Limit Theorems for Difference of Chi-type processesAug 11 2015Jul 15 2016Let $\{\zeta_{m,k}^{(\kappa)}(t), t \ge0\}, \kappa>0$ be random processes defined as the differences of two independent stationary chi-type processes with $m$ and $k$ degrees of freedom. In applications such as physical sciences and engineering dealing ... More

Comparison Inequalities for Order Statistics of Gaussian ArraysMar 31 2015Normal comparison lemma and Slepian's inequality are essential tools in the study of Gaussian processes. In this paper we extend normal comparison lemma and derive various related comparison inequalities including Slepian's inequality for order statistics ... More