  In this paper, we derive the joint asymptotic distributions of functions of
quantile estimators (the non-parametric sample quantile and the parametric
location-scale quantile estimator) with functions of measure of dispersion
estimators (the sample variance, sample mean absolute deviation, sample median
absolute deviation) - assuming an underlying identically and independently
distributed sample. We also discuss the conditions required by the use of such
estimators. Further, we show that these results can be extended to any higher
order absolute central sample moment as measure of dispersion. Aware of the
difference in speed of convergence of the two quantile estimators, we compare
the impact of the choice of the quantile estimator (and measure of dispersion)
on the asymptotic correlations. Then we prove a scaling law for the asymptotic
dependence of quantile estimators with measure of dispersion estimators.
Finally, we show a good finite sample performance of the asymptotics in
simulations for elliptical distributions. All the results should constitute an
important and useful complement in the statistical literature as those
estimators are either of standard use in statistics and application fields, or
should become as such because of weaker conditions in the asymptotic theorems.
