The Horizon of a Random Cone Field under a Trend: One-Dimensional Distributions

The horizon $\xi_T(x)$ of a random field $\zeta(x,y)$ of right circular cones on a plane is investigated. It is assumed that bases of cones are centered at points $s_n=(x_n,y_n)$, $n=1,2,\dots$, on the $(X,Y)$, constituting a Poisson point process $S$ with intensity $\lambda_0>0$ in a strip $\Pi_T=\{(x,y):-\infty<x<\infty$, $0\le y\le T\}$, while altitudes of the cones $h_1,h_2,\dots$ are of the form $h_n=h_n^\ast+f(y_n)$, $n=1,2,\dots$, where $f(y)$ is an increasing continuous function on $[0,\infty)$, $f(0)=0$, and $h_1^*,h_2^*,\dots$ is a sequence of independent identically distributed positive random variables, which are independent of the Poisson process $S$ and have a distribution function $F(h)$ with density $p(h)$.
For some choices of the distribution function $F(h)$ and the trend function $f(y)$, limiting one-dimensional distributions (as $T\to\infty$) of the process $\xi_T(x)$ are obtained.