Some bounds for the probabilities of events generated by Gaussian processes and fields, and applications to crossings of a level

Lower bounds are obtained for the probabilities of events generated by a Gaussian family $x(t)$, $t\in[0,T]$ or $x(t_1,\dots,t_m)$, $t_1^2+\dots+t_m^2\le T^2$, as $T\to\infty$.