Numbers of near-maxima for the bivariate case
Let \((X_i,Y_i)\), \(1\leq i\leq n\), be independent and identically distributed random vectors with a continuous distribution function. Put \(X_{n:n}:=\max_{1\leq i\leq n}X_i\) and denote by \(Y_{(n):n}\) the concomitant associated with \(X_{n:n}\). Denote by \(K_n(a,b_1,b_2)\) the number of sample elements that belong to the open rectangle \((X_{n:n}-a,X_{n:n})\times (Y_{(n):n}-b_1,Y_{(n):n}-b_2)\). The authors establish in particular convergence in probability and weak convergence of \(K_n(a,b_1,b_2)\), suitably standardized, as \(n\to\infty\).
scientific article; zbMATH DE number 5666077
scientific article