The limit theorems for maxima of stationary Gaussian processes with random Index

Let {X(t), t a parts per thousand yen 0} be a standard (zero-mean, unit-variance) stationary Gaussian process with correlation function r(center dot) and continuous sample paths. In this paper, we consider the maxima M(T) = max{X(t), is not an element of t a [0, T]} with random index T (T) , where T (T) /T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M(T (T) ) exists under some additional conditions related to the correlation function r(center dot).