Limit laws for the cumulative number of ties for the maximum in a random sequence

Let {Xn,n⩾1} be a sequence of independent identically distributed random variables, taking nonnegative integer values. An observation Xn is a tie for the maximum if Xn=max{X1,…,Xn-1}. In this paper, we obtain weak and strong laws of large numbers and central limit theorems for the cumulative number of ties for the maximum among the first n observations.