Abstract
Let {Xn, n≥1} be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call Xn a δ-record if Xn>max {X1, …, Xn−1}+δ, where δ is an integer constant. We use martingale arguments to show that the counting process of δ-records among the first n observations, suitably centered and scaled, is asymptotically normally distributed for δ≠0. In particular, taking δ=−1 we obtain a central limit theorem for the number of weak records.
Citation
Raúl Gouet. F. Javier López. Gerardo Sanz. "Asymptotic normality for the counting process of weak records and δ-records in discrete models." Bernoulli 13 (3) 754 - 781, August 2007. https://doi.org/10.3150/07-BEJ6027
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