# zbMATH — the first resource for mathematics

The stochastic process of segments on the two-dimensional Euclidean sphere. II. (Russian) Zbl 0696.60016
This work continues the authors’ researches beginning in ibid. 39, 107- 113 (1988; Zbl 0665.60019), and devoted to the further study of the properties of the sphere covered by random caps [see R. E. Miles, Sankhyā, Ser. A 33, 145-174 (1971; Zbl 0243.60014); P. A. P. Moran and S. Fazekas de St. Groth, Biometrika 49, 389-396 (1962; Zbl 0108.316)].
We investigate a random cap process $${\mathcal A}$$ on the two-dimensional Euclidean sphere $$S^ 2$$ of unit radius. Each trajectory of the process $${\mathcal A}$$ is an unordered set $$\{Q_{(i)}(U_{(i)},a_{(i)})\}$$ consisting of mutually disjoint hemispherical caps. A position of any cap $$Q_{(i)}(U_{(i)},a_{(i)})$$ on $$S^ 2$$ is identically determined by the pair $$[U_{(i)}(\phi_{(i)}, \theta_{(i)});a_{(i)}]$$ where $$(\phi_{(i)},\theta_{(i)})$$ are the spherical coordinates of the random cap center $$U_{(i)}$$. Cap diameters $$\{a_{(i)}\}$$ take values from the general population $$K=[0,A]$$ where $$A<\pi$$ with probability density f(a) and N is a nonnegative integer random variable. The process $${\mathcal A}$$ is considered as a random unordered marked point process (MPP) ($${\mathcal E}^*_{{\mathcal A}},{\mathcal X}^*_{{\mathcal A}},P^*_{{\mathcal A}})$$ with the trajectories $$E^*_{{\mathcal A}}= \{[U_{(i)}; a_{(i)}]\}\in {\mathcal E}^*_{{\mathcal A}}$$ in the bounded space $$(Y,U_ y,B_ y)$$ where $$Y=S^ 2\times K$$, $$U_ y=U_{S^ 2}\otimes U_ k$$, $$B_ y=B_{S^ 2}\odot B_ k$$ [see the authors, loc. cit.]. We can put in correspondence a random unordered MPP of parameters $${\mathcal D}=({\mathcal E}^*_{{\mathcal D}},{\mathcal X}^*_{{\mathcal D}},P^*_{{\mathcal D}})$$ with the trajectories $$E^*_{{\mathcal D}}=\{[\phi_{(i)}, \theta_{(i)};a_{(i)}]\}$$ in the bounded space $$(Z,U_ z,B_ z)$$ where $$Z=\Delta_{\phi,\theta}\times K$$ $$(\Delta_{\phi,\theta}=\{(\phi,\theta):$$ $$0\leq \phi <2\pi$$, $$-\pi /2<\theta <\pi /2\}\cup \{(0,\pi /2),(0,-\pi /2)\})$$ to the process $${\mathcal A}.$$
We shall propose that the processes $${\mathcal A}$$ and $${\mathcal D}$$ possess the following properties:
1. The random variable N has a finite expectation: $$E[N]<\infty.$$
2. The point process (PP) $$\tilde {\mathcal A}=({\mathcal E}_{\tilde {\mathcal A}},U_{\tilde {\mathcal A}},P_{\tilde {\mathcal A}})$$ of the random cap centers has a constant intensity $$\lambda$$.
3. The MPP $${\mathcal D}$$ is a random unordered simple PP with independent marking in the bounded space $$(Z,U_ z,B_ z)$$ [see the authors, loc. cit.].
Theorem. For any $$\bar Z\in U_ z$$ a moment measure of the first order $$O^{(1)}(\bar Z)=E [N^*(E^*_ D,\bar Z)]$$ of the process $${\mathcal D}$$ is calculated by the formula $O^{(1)}(\bar Z)=\iiint_{(\phi,\theta,a)\in \bar Z}\lambda \cos \theta \quad f(a)d\phi d\theta da$ where $$N^*(E^*_{{\mathcal D}},\bar Z)=card[E^*_{{\mathcal D}}\cap \bar Z]$$.
Reviewer: Yu.I.Petunin

##### MSC:
 60D05 Geometric probability and stochastic geometry 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)