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

Authors:	Raúl Gouet F Javier López Gerardo Sanz

Institution:	1. Departmento de Ingeniería Matemática and Centro de Modelamiento Matemático (CNRS UMI 2807), Universidad de Chile, Av. Blanco Encalada 2120, 837-0459 Santiago, Chile;2. Departmento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, C/ Pedro Cerbuna, 12, 50009 Zaragoza, Spain

Abstract:	Let {_X_n,n?1} ${X_{n}, n ? 1}$ be a sequence of independent identically distributed random variables, taking nonnegative integer values. An observation _X_n $X_{n}$ is a tie for the maximum if _X_n=max{_X₁,…,_X_n_-₁} $X_{n} = \max {X_{1}, \dots, X_{n - 1}}$ . 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 $n$ observations.

Keywords:	Ties Records Weak records Discrete distributions Limit theorems
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