The stochastic process of segments on the two-dimensional Euclidean sphere. II. (Russian) Zbl 0696.60016

Petunin, Yu. I.; Semejko, N. G.

Teor. Veroyatn. Mat. Stat., Kiev 41, 88-96 (1989).

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)

Keywords:

properties of the sphere covered by random caps; random cap process; point process

Citations:

Zbl 0665.60019; Zbl 0243.60014; Zbl 0108.316