Asymptotic normality for the counting process of weak records and δ-records in discrete models

Details Asymptotic normality for the counting process of weak records and δ-records in discrete models Asymptotic normality for the counting process of weak records and δ-records in discrete models

2007-09-05

arxiv.org/abs/0709.0620v1

Bernoulli 2007, Vol. 13, No. 3, 754-781

Mathematics

Probability

Published at http://dx.doi.org/10.3150/07-BEJ6027 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statist

Scientific paper

10.3150/07-BEJ6027

Let $\{X_n,n\ge1\}$ be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call $X_n$ a $\delta$-record if $X_n>\max\{X_1,...,X_{n-1}\}+\delta$, where $\delta$ is an integer constant. We use martingale arguments to show that the counting process of $\delta$-records among the first $n$ observations, suitably centered and scaled, is asymptotically normally distributed for $\delta\ne0$. In particular, taking $\delta=-1$ we obtain a central limit theorem for the number of weak records.

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