Introduction to Quantiles

Let $F$ be a CDF. From Analysis we know that:

• If $F$ is continous and strictly monotonic then $F(x)=p$ has $F^{-1}(p)$ as solution
• If $F$ is only continous then the solution is a nonempty compact interval (more than one solution)
• If $F$ does not attain value $p$ then there is a jump above it

Let $F$ be a CDF which is the limit of a sequence of ECDFS.

• If $F$ has exactly one p-quantile then the smallest and largest quantiles of the sequence of ECDFs converge to this p-quantile
• Every Accumulation Point of the smallest and largest quantiles of the sequence of CDFs is a p-quantile of $F$