  Let $(X,Y)$ be a bivariate random vector. The estimation of a probability of
the form $P(Y\leq y \mid X >t) $ is challenging when $t$ is large, and a
fruitful approach consists in studying, if it exists, the limiting conditional
distribution of the random vector $(X,Y)$, suitably normalized, given that $X$
is large. There already exists a wide literature on bivariate models for which
this limiting distribution exists. In this paper, a statistical analysis of
this problem is done. Estimators of the limiting distribution (which is assumed
to exist) and the normalizing functions are provided, as well as an estimator
of the conditional quantile function when the conditioning event is extreme.
Consistency of the estimators is proved and a functional central limit theorem
for the estimator of the limiting distribution is obtained. The small sample
behavior of the estimator of the conditional quantile function is illustrated
through simulations.
