The Horizon of the Random Cone Field Under a Trend

The horizon . T(x) of a random field . (x, y) of right circular cones on a plane is investigated. It is supposed that bases of cones are centered at points sn = (xn, yn), n = 1, 2, ···, on the (X, Y)-plane, constituting a Poisson point process S with intensity .0 > 0 in a strip .T = {(x, y): . .< x <., 0 . y . T}, while altitudes of the cones h1, h2, · ·· are of the form hn = hn* + f(yn), n = 1, 2, ···, where f(y) is an increasing continuous function on [0,.), f(0) = 0, and h1*, h2*, · ·· is a sequence of i.i.d. positive random variables which are independent of the Poisson process S and have a distribution function F(h) with density p(h). Let denote the expected mean number of local maxima of the process . T(x) per unit length of the X-axis. We obtain an exact formula for under an arbitrary trend function f(y). Conditions sufficient for the limit being infinite are obtained in two cases: (a) h1* has the uniform distribution in [0, H], f(y) = ky.; (b) h1* has the Rayleigh distribution, f(y) = k[log(y + 1)].. (In both cases . 0 and 0 < k..) The corresponding sufficient conditions are: 0 < .< 1 in case (a), 0 < .< 1/2 in case (b).