Numbers of near-maxima for the bivariate case

Let Z¯1=(X1,Y1)…,Z¯n=(Xn,Yn) be independent and identically distributed random vectors with continuous distribution. Let Kn(a,b1,b2) be the number of sample elements that belong to the open rectangle (Xmax(n)−a,Xmax(n))×(Ymax(n)−b1,Ymax(n)+b2) - numbers of near-maxima in the bivariate case. In the present paper, we discuss asymptotic properties of Kn(a,b1,b2) and Kn(∞,0,∞).