Numbers of near bivariate record-concomitant observations

Abstract

Let (Z) over bar (1) = (X(1), Y(1)), (Z) over bar (2) = (X(2), Y(2)), ... be independent and identically distributed random vectors with continuous distribution. Let L(n) and X (n) denote the nth record time and the nth record value obtained from the sequence of Xs. Let Y(n) denote the concomitant of the nth record value, which relates to the sequence of Ys. We call (Z) over bar (i) a near bivariate nth record-concomitant observation if (Z) over bar (i) belongs to the open rectangle (X (n) - a, X (n)) x (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. (c) 2011 Elsevier Inc. All rights reserved.