On limit distributions of first crossing points of Gaussian sequences

Let {Xk, k[epsilon]Z} be a stationary Gaussian sequence with EX1 - 0, EX2k = 1 and EX0Xk = rk. Define [tau]x = inf{k: Xk >- [beta]k} the first crossing point of the Gaussian sequence with the function - [beta]t ([beta] > 0). We consider limit distributions of [tau]x as [beta]-->0, depending on the correlation function rk. We generalize the results for crossing points [tau]x = inf{k: Xk >[beta][finite part integral](k)} with [finite part integral](- t)[approximate, equals]t[gamma]L(t) for t-->[infinity], where [gamma] > 0 and L(t) varies slowly.