| Summary: | 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.
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