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

Let { X n , n ⩾ 1 } be a sequence of independent identically distributed random variables, taking nonnegative integer values. An observation X n is a tie for the maximum if X n = max { X 1 , … , X n - 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.