The stochastic process of segments on the two-dimensional Euclidean sphere. I.

*(Russian)*Zbl 0665.60019This work is the first part of research devoted to the construction and investigation of the properties of the cap random process U on the two- dimensional Euclidean sphere. There are many applied sciences such as astronomy, cosmology, selenography, cytology, virusology in which further progress is impossible without a mathematical model of the cap process on the sphere \(S^ 2\). The process U is considered as a simple unordered marked point process (MPP) in the phase space \(S^ 2\times K\) where \(S^ 2\) is the space of the cap centers (position points), \(K=[0,A]\) is the space of cap diameters (marks) \((A<<\pi).\)

A simple MPP is represented by J. E. Moyal’s conception [see Acta Math. 108, 1-31 (1962; Zbl 0128.403)]. By using the methods of the theory of order statistics we introduce the concept of a simple unordered MPP with independent marking (PPIM) with the help of the theorem on renewal samples. From our point of view, the well-known A. Prekopa [see Ann. Univ. Sci. Budapest, Rolando Eötvös, Sect. Mat. 1, 153-170 (1958; Zbl 0089.340)] definition of the PPIM is insufficient for applications. We construct a point process of the position points by using a projection of MPP on the position space.

A simple MPP is represented by J. E. Moyal’s conception [see Acta Math. 108, 1-31 (1962; Zbl 0128.403)]. By using the methods of the theory of order statistics we introduce the concept of a simple unordered MPP with independent marking (PPIM) with the help of the theorem on renewal samples. From our point of view, the well-known A. Prekopa [see Ann. Univ. Sci. Budapest, Rolando Eötvös, Sect. Mat. 1, 153-170 (1958; Zbl 0089.340)] definition of the PPIM is insufficient for applications. We construct a point process of the position points by using a projection of MPP on the position space.

Reviewer: Y.I.Petunin

##### MSC:

60D05 | Geometric probability and stochastic geometry |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |