The number and sum of near-maxima for thin-tailed populations (Q2713160)

For a given sample \( X_1,\dots,X_n \) of iid rvs with df \( F \) we call \( X_j \) a near-maximum (observation) if NEWLINE\[NEWLINE\max_{1 \leq k \leq n} X_k -a < X_j \leq \max_{1 \leq k \leq n} X_k =: M_nNEWLINE\]NEWLINE with fixed \( a>0 \). The investigation of the number of near-maxima NEWLINE\[NEWLINEK_n(a) = \sum_j I{X_j \in (M_n-a, M_n]}NEWLINE\]NEWLINE and its asymptotic behaviour was started by the author and \textit{F. W. Steutel} [Aust. J. Stat. 39, No. 2, 179-192 (1997; Zbl 0908.60040)] and studied further by the author and \textit{Y. Li} [Stat. Probab. Lett. 40, No. 4, 395-401 (1998; Zbl 0935.60014) and Commun. Stat., Theory Methods 27, No. 3, 673-686 (1998; Zbl 0903.60044)], \textit{Y. Li} (1999), \textit{E. Hashorva} and \textit{J. Hüsler} [J. Multivariate Anal. 68, No. 2, 212-225 (1999; Zbl 0921.60041)], \textit{Y. Li} and the author (2000). Their results affirm the picture of the general asymptotic behaviour of the maxima: \( M_n \) is isolated from all other observations when \( F \) is thick; there is only a finite number of near-maxima when \( F \) has a medium tail; and the number of near-maxima grows to infinity when \( F \) has a thin tail. NEWLINENEWLINENEWLINEThe present paper completes the picture in giving limit results for \( K_n(a) \) in the case of thin tailed \( F \). Then the (nonlinear) normalization depends on how thin the tail is: svelte, lean or graunt. The same methodology works for the number \( K_n(a,m) \) of near \(m\)-extremes and the asymptotic behaviour of their sum. It is shown that the LLN and the CLT are not affected by deleting near-maxima from the sample.