This article is cited in
3 papers
Random time-changed extremal processes
E. I. Panchevaa,
E. T. Kolkovskab,
P. K. Jordanovac a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
b Center for Mathematical Research
c Konstantin Preslavsky University of Shumen
Abstract:
The point process
$\mathcal N=\{(T_k,X_k):k\ge 1\}$ we deal with here is an assumed Bernoulli point process with independent random vectors
$X_k$ in
$[0,\infty)^d$ and with random time points
$T_k$ in
$[0,\infty)$, independent of
$X$. For normalizing we use a regular sequence
$\xi_n(t,x) =(\tau_n(t),u_n(x))$ of time-space changes of
$[0,\infty)^{1+d}$. We consider the sequence of the associated extremal processes, $\widetilde{Y}_n(t)=\{\bigvee u^{-1}_n(X_k):T_k\le\tau_n(t)\}$, where the max-operation "
$\vee$" is defined in
$\mathbf R^d$ componentwise. We assume further that there exist a stochastically continuous time process
$\theta=\{\theta(t):t\ge 0\}$, strictly increasing and independent of
$\{X_k\}$, and an integer-valued deterministic counting function
$k$ on
$[0,\infty)$, so that the counting process
$N$ of
$\mathcal N$ has the form
$N(s)=k(\theta(s))$ a.s. In this framework we prove a functional transfer theorem which claims in general that if $\tau_n^{-1}\circ\theta\circ\tau_n\Rightarrow\Lambda$, where
$\Lambda$ is strictly increasing and stochastically continuous, and if $\bigvee_{k=1}^{k(\tau_n(\cdot))}u^{-1}_n (X_k)\Rightarrow Y(\cdot)$, then $\widetilde{Y}_n\rightarrow\widetilde{Y}=Y\circ\Lambda$, where
$Y$ is a self-similar extremal process. We call such limit processes random time-changed, or compound. They are stochastically continuous and self-similar with respect to the same one-parameter norming group as
$Y$. We show that the compound process is an extremal process (i.e., a process with independent max-increments) if and only if
$\Lambda$ has independent increments and
$Y$ has homogeneous max-increments. We apply random time-changed extremal processes to find a lower bound for the ruin probability in an insurance model associated with
$\mathcal N$. We give also an upper bound using an
$\alpha$-stable Lévy motion.
Keywords:
extremal processes, weak limit theorems, ruin probability. Received: 11.08.2003
Revised: 15.04.2005
Language: English
DOI:
10.4213/tvp23
Fulltext:
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