I think the title is self-explanatory. I understand that the skewness and the tail behavior of some distribution are completely unrelated as any symmetric distribution will have a skewness of zero irrespective of how heavy its tails are.
However, I was wondering (i) if the skew-normal distribution and the skew-Cauchy distribution are heavy-tailed.
(ii) Also, we can get skewed distribution using a selection procedure. That is, if $X$ and $Y$ are Gaussian (or Cauchy), $\frac{d}{dz}P(X-kY\leq z|Y>0)$ represents a density of a skewed version of normal (or Cauchy distribution). What about the tails of such constructed skewed distributions?
Note: I say the distribution of a random variable $X$ heavy-tailed if $\lim \limits_{x\rightarrow \infty} e^{tx} P(X>x) = \infty$ for all $t>0$.