Asymptotic distribution of sum and maximum for Gaussian processes

Let { X n , n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r ( n ) = E X 0 X n . Let Under some mild regularity conditions on r ( n ) and the condition that r ( n )ln n = o (1) or ( r ( n )ln n ) −1 = O (1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.