On the sequence of partial maxima of some random sequences

Let {Xn, n [greater-or-equal, slanted] 1} be a sequence of identically distributed random variables, Zn = max {X1,..., Xn} and {un, n [greater-or-equal, slanted] 1 } an increasing sequence of real numbers. Under certain additional requirements, necessary and sufficient conditions are given to have, with probability one, an infinite number of crossings of {Zn} with respect to {un}, in two cases: (1) The Xn's are independent, (2) {Xn} is stationary Gaussian and satisfies a mixing condition.