A limit theorem for intersections of random semi-spaces. (English. Ukrainian original) Zbl 0861.60056

Molchanov, I. S.

Theory Probab. Math. Stat. 49, 147-150 (1994); translation from Teor. Jmovirn. Mat. Stat. 49, 207-211 (1993).

Summary: A limit theorem is proved for multiplicatively normalized intersections of random semi-spaces of the form \(\{{\mathbf x}\in{\mathbf R}^d:{\mathbf x}\cdot\xi_i\leq |\xi_i|^2\}\), where \(\xi_i\), \(i\geq 1\), are independent identically distributed random vectors. For the limit closed random set \({\mathbf X}\) and any compact set \({\mathbf K}\) the probability \(P\{{\mathbf K}\subset{\mathbf X}\}\) and the mathematical expectation of the volume are found.

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60F15	Strong limit theorems
60B10	Convergence of probability measures

Keywords:

multiplicatively normalized intersections of random semi-spaces; random set