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

Let {X(t), t ≥ 0} be a standard (zero-mean, unit-variance) stationary Gaussian process with correlation function r(·) and continuous sample paths. In this paper, we consider the maxima M(T) = max{X(t), ∀t ∈ [0, T]} with random index TT, where TT/T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M(TT) exists under some additional conditions related to the correlation function r(·).