Conditions of absence of singular points on the level set of a random field. (Russian) Zbl 0632.60046

Radchenko, A. N.

Teor. Veroyatn. Mat. Stat., Kiev 36, 114-119 (1987).

Information about the absence of critical points on a given level set of random processes or fields is necessary in problems of level crossings. A well-known example of such results is Bulinskaya’s theorem. This paper presents sufficient conditions for the field \[ \xi: G\subset {\mathbb{R}}^ m\to {\mathbb{R}}^ n,\quad \xi \in C^ 1(G)\quad a.s. \] (in terms of local behaviour of \(\xi\) and \(\partial \xi_ i/\partial t_ j)\) for \[ P(\exists \tau \in G: \xi (\tau)=0,\quad f(\partial \xi_ i(\tau)/\partial t_ j)=0)=0, \] where \(f: {\mathbb{R}}^{mn}\to {\mathbb{R}}^ 1\) is a nonrandom continuous function.

MSC:

60G60

Random fields

Keywords:

critical points; level set; problems of level crossings; local behaviour