Geodesic
Weak convergence of the horyzon for the random field of cones in the expanding strip
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 4, pp. 693-706 
V. P. Nosko. Weak convergence of the horyzon for the random field of cones in the expanding strip. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 4, pp. 693-706. http://geodesic.mathdoc.fr/item/TVP_1982_27_4_a5/
@article{TVP_1982_27_4_a5,
     author = {V. P. Nosko},
     title = {Weak convergence of the horyzon for the random field of cones in the expanding strip},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {693--706},
     year = {1982},
     volume = {27},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_4_a5/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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. 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.