A note on extremes of concomitants of order statistics
For a sequence \((X_n,Y_n)_{n\geq 1}\) of identically distributed (not necessarily independent) bivariate r.v.s an asymptotic distribution \(M_{k:n}=\max(Y_{[n-k-1:n]},\dots,Y_{[n:n]})\) is studied (\(n\to\infty\)), where \(Y_{[r:n]}+\sum_{i=1}^n Y_i{\mathbf 1}\{X_{r:n}=X_i\}\) is the cocomitant of the \(r\)-th order statistics \(X_{r:n}\). It is shown that under some conditions the asymptotic distribution of \(M_{n:k}\) coincides with the asymptotic distribution of i.i.d. \((X_i,Y_i)\). These conditions include the asymptotic conditional independence of \(Y_1,\dots, Y_n\) for fixed \(X_1,\dots,X_n\) and the condition \(A_1(u):\) \[ \sup_{j_i,x_i}|\Pr\{Y_{j_0}<u_n |X_{j_0}=x_{j_0}\} -\Pr\{Y_{j_i}<u_n |X_{j_i}=x_{j_i}, i=0,1,\dots,p\}|=o(1) \] for some real sequence \(u_n\).
scientific article; zbMATH DE number 1833855