Weak convergence of the horyzon for the random field of cones in the expanding strip

V. P. Nosko

Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 4, pp. 693-706

Résumé

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.