Limit laws for the cumulative number of ties for the maximum in a random sequence (Q2390459)

Let \(\{X_n,n \geq 1\}\) be a sequence of independent, identically distributed, nonnegative integer-valued random variables. An observation \(X_n\) is said to be a tie for the maximum if \(X_n=\max \{X_1,\ldots,X_{n-1}\}\). The counting process of ties is denoted by \(T_n=\sum_{i=1}^n I_i, n \geq 1\), where \(I_i=\boldsymbol1_{\{X_i=M_{i-1}\}}\) is the indicator of a tie. The authors obtain weak and strong laws of large numbers and central limit theorems for the process \(T_n\).