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Weak convergence of the horyzon for the random field of cones in the expanding strip
V. P. Nosko Moscow
Abstract:
We consider a random process
$\xi_T(x)$ which can be interpreted as the horyzon of the random field
$z=\zeta(x,y)$ generated by circular cones which are placed at random in the strip
$\{(x,y)\colon-\infty<x<\infty,\,0\le y\le T\}$. It is assumed, that the heights of cones are non-negative i. i. d. random variables with distribution function
$F(h)$. Vertex angles of the cones are assumed to be equal.
It is shown that there exist non-random positive functions
$f(T)$ and
$g(T)$ such that the transformed process
$$
\widetilde\xi_T(x)=g(T)[\xi_T\biggl(\frac{x}{g(T)}\biggr)-f(T)]
$$
converges in distribution (when
$T\to\infty$) to a continuous random process
$\widetilde\xi_\infty(x)$ whose
finite-dimensional distributions are given in a closed form. In accordance with the type of
$F(h)$, one-dimensional distributions of
$\widetilde\xi_\infty(x)$ are the limiting distributions of the extremal
values.
Received: 15.12.1980
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