Numbers of near bivariate record-concomitant observations

Let Z¯1=(X1,Y1),Z¯2=(X2,Y2),… be independent and identically distributed random vectors with continuous distribution. Let L(n)L(n) and X(n)X(n) denote the nnth record time and the nnth record value obtained from the sequence of XXs. Let Y(n)Y(n) denote the concomitant of the nnth record value, which relates to the sequence of YYs. We call Z¯i a near bivariate nnth record-concomitant observation if Z¯i belongs to the open rectangle (X(n)−a,X(n))×(Y(n)−b1,Y(n)+b2)(X(n)−a,X(n))×(Y(n)−b1,Y(n)+b2), where a,b1,b2>0a,b1,b2>0 and L(n)