A note on the number of records near the maximum
Continuing work by \textit{A. G. Pakes} and the reviewer [Aust. J. Stat. 39, No. 2, 179-192 (1997; Zbl 0908.60040)], the author considers the quantity \(K_n(a)\), the number of values within a distance \(a\) from the maximum in a random sample of size \(n\). It is proved that \(EK_n(a)\to 1\) if and only if \(K_n(a)\to 1\) in probability. A necesary and sufficient condition for this in terms of tail behaviour is given, and a sufficient condition for \(K_n(a)\to 1\) almost surely.
scientific article; zbMATH DE number 1310366