A limit formula and a series expansion for the bivariate Normal tail probability

Abstract

This work presents a limit formula for the bivariate Normal tail probability. It only requires the larger threshold to grow indefinitely, but otherwise has no restrictions on how the thresholds grow. The correlation parameter can change and possibly depend on the thresholds. The formula is applicable regardless of Salvage’s condition. Asymptotically, it reduces to Ruben’s formula and Hashorva’s formula under the corresponding conditions, and therefore can be considered a generalisation. Under a mild condition, it satisfies Plackett’s identity on the derivative with respect to the correlation parameter. Motivated by the limit formula, a series expansion is also obtained for the exact tail probability using derivatives of the univariate Mill’s ratio. Under similar conditions for the limit formula, the series converges and its truncated approximation has a small remainder term for large thresholds. To take advantage of this, a simple procedure is developed for the general case by remapping the parameters so that they satisfy the conditions. Examples are presented to illustrate the theoretical findings.