Asymptotic properties of the point processes of rare events in Gaussian fields. I

The asymptotic behaviour of point processes, generated by arbitrary rare events defined on the stationary discrete Gaussian vector fields, is studied. This paper contains some sufficient conditions for the weak convergence (after a suitable change of a parameter space) of the rare events point processes to a Poisson process. These conditions are weaker than well-known conditions for the special case of high level crossings.