  For a fixed positive integer $\;k,\;$ limit laws of linearly normalized
$\;k$-th upper order statistics are well known. In this article, a
comprehensive study of tail behaviours of limit laws of normalized $k$-th upper
order statistics under fixed and random sample sizes is carried out using tail
equivalence which leads to some interesting tail behaviours of the limit laws.
These lead to definitive answers about their max domains of attraction.
Stochastic ordering properties of the limit laws are also studied. The results
obtained are not dependent on linear norming and apply to power norming also
and generalize some results already available in the literature. And the proofs
given here are elementary.
