An asymptotic relation for the expected number of excursions above a boundary g(n) by a random walk S n , n = 1,2, .., N is given in terms of an integral involving g. An integral test is given to determine whether the total excursion time has finite expectation. If some moment assumptions hold then the expectation of the total excursions is finite if and only if J°° t 1 * 2 g~1{t)exp (-g 2 (t)/2 > )dt < oo.