Numbers of near bivariate record-concomitant observations

Let Z ¯ 1 = ( X 1 , Y 1 ) , Z ¯ 2 = ( X 2 , Y 2 ) , … be independent and identically distributed random vectors with continuous distribution. Let L ( n ) and X ( n ) denote the n th record time and the n th record value obtained from the sequence of X s. Let Y ( n ) denote the concomitant of the n th record value, which relates to the sequence of Y s. We call Z ¯ i a near bivariate n th record-concomitant observation if Z ¯ i belongs to the open rectangle ( X ( n ) − a , X ( n ) ) × ( Y ( n ) − b 1 , Y ( n ) + b 2 ) , where a , b 1 , b 2 > 0 and L ( n ) < i < L ( n + 1 ) . Asymptotic properties of the numbers of near bivariate record-concomitant observations are discussed in the present work. New techniques for generating bivariate record-concomitants, the numbers of near record observations and the numbers of near bivariate record-concomitant observations are also proposed.