186 research outputs found
On the asymptotic distribution of certain bivariate reinsurance treaties
Let (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 asymptotic distribution of randomly indexed
order statistics of the random sample
(X_1,Y_1),(X_2,Y_2),...,(X_{N(t)},Y_{N(t)}) which is then used to obtain
asymptotic representations for the joint distribution of two generalised
largest claims reinsurance treaties available under specific insurance
settings. As a by-product we obtain a stochastic representation of a
m-dimensional Lambda-extremal variate in terms of iid unit exponential random
variables.Comment: 11 page
Extremes of Aggregated Dirichlet Risks
The 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 has df in the Gumbel or the Weibull max-domain of
attraction. We present further results for the joint asymptotic independence
and the max-sum equivalence.Comment: published versio
Conditional Limit Results for Type I Polar Distributions
Let (S_1,S_2)=(R \cos(\Theta), R \sin (\Theta)) be a bivariate random vector
with associated random radius R which has distribution function being
further independent of the random angle \Theta. In this paper we investigate
the asymptotic behaviour of the conditional survivor probability
\Psi_{\rho,u}(y):=\pk{\rho S_1+ \sqrt{1- \rho^2} S_2> y \lvert S_1> u}, \rho
\in (-1,1),\in R when u approaches the upper endpoint of F. On the density
function of \Theta we require a certain local asymptotic behaviour at 0,
whereas for F we require that it belongs to the Gumbel max-domain of
attraction. The main result of this contribution is an asymptotic expansion of
\Psi_{\rho,u}, which is then utilised to construct two estimators for the
conditional distribution function 1- \Psi_{\rho,u}. Further, we allow \Theta to
depend on u.Comment: 14 pages, paper submitted to Extremes in 200
Piterbarg Theorems for Chi-processes with Trend
Let be a chi-process with
degrees of freedom where 's are independent copies of some generic
centered Gaussian process . This paper derives the exact asymptotic behavior
of P{\sup_{t\in[0,T]} \chi_n(t)>u} as u \to \infty, where is a given
positive constant, and is some non-negative bounded measurable
function. The case is investigated in numerous contributions by
V.I. Piterbarg. Our novel asymptotic results for both stationary and
non-stationary are referred to as Piterbarg theorems for chi-processes with
trend.Comment: 22 page
Tail Behaviour of Weighted Sums of Order Statistics of Dependent Risks
Let be 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 of under the general case in which the
distribution function of is long-tailed or rapidly varying and may not be comparable in terms of their tail probability.
We also present two examples and an application of our results in risk theory
Boundary Non-crossings of Brownian Pillow
Let B 0(s,t) be a Brownian pillow with continuous sample paths, and let h,u:[0,1]2→ℝ be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability Further we investigate the asymptotic behaviour of ψ(u;γ h) with γ tending to ∞ and solve a related minimisation proble
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