On the number of near-maxima (Q2704105)

The authors consider the number \(K_{n}(a,b)\) of `near-maxima', i.e., the number of points in a random sample \(X_{1}, \dots,X_{n}\) within an interval \([a,b)\) from the sample maximum \(M_{n}\): \(K_{n}(a,b)=\sum_{i=0}^{n}1_{\{a \leq M_{n}-X_{i}<b\}}\), as considered by \textit{A. G. Pakes} and \textit{F. W. Steutel} [Aust. J. Stat. 39, No. 2, 179-192 (1997; Zbl 0908.60040)]. They extend and generalize some of these results. Especially, for \(0<a_{n}<b_{n}<a_{n}'<b_{n}'\) they consider the joint distribution of \(K_{n}(a_{n},b_{n})\) and \(K_{n}(a_{n}',b_{n}')\) and obtain its limit behaviour. They show that under suitable conditions the point process induced by \(K_{n}(a_{n},b_{n})\) converges to a Cox process. For specific choices of the underlying distribution they obtain explicit results for the rate of convergence or even the limit of \(EK_{n}(0,a_{n})\) as \(n\to \infty\). NEWLINENEWLINENEWLINEReviewer's remark: It would seem that the equality sign in formula (19) should be an `asymptotically-equal' sign.NEWLINENEWLINENEWLINE\textbf{Shigeo Takenaka (Okayama)}: Let \(\{X_n\}\) be i.i.d. random variables with common continuous distributionfunction \(F\). Define the maximum \(M_n=\max\{X_i;i\leq n\}\) and the number of near-maxima \(K_n(a,b)=\sum_{i=1}^{n} 1_{\{a\leq M_n - X_i < b\}}\). The generating function of \(K_n(a,b)\) is calculated as\({\mathbf E}(s^{K_n(a.b)})=n\int(F(x)-(1-s)[F(x-a)-F(x-b)])^{n-1}dF(x)\). Further, assume that there exists sequences \(q_n,\, r_n\), such that\(F^n(q_nx+r_n)\rightarrow H(x),\;n\rightarrow\infty\), then the asymptotic joint generation function is obtained as \({\mathbf E}(s^{K_n(0,b_n)}t^{K_n(a'_n.b'_n)}) \rightarrow s\int_{\alpha(H)+c_{b'}}^\infty\left({H(x-c_b)} \over {H(x)}\right)^{1-s}\left({H(c-c_{b'})} \over {H(x-c_{a'})}\right)^{1-t} dH(x)\), where \(\alpha(H)=sup\{x;H(x)=0\}\) and \(\lim {{*_n}\over{q_n}} = c_{*}\).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00040].