Are the skew-normal distribution and the skew-Cauchy distribution heavy-tailed? 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 $t>0$.
 A: I'll take it that you're happy to take it as a given that the ordinary (symmetric) normal is light tailed and the ordinary (symmetric) Cauchy is heavy tailed.
That the indicated skew-normal is not heavy-tailed is easy to see. I'll discuss the 'standard case' of the skew-normal; the scale-location case is not really any more work.
Taking $f(x)=2\phi (x)\Phi (\alpha x)$, we can immediately see that $f(x)<2\phi(x)$, so 
 $P(X>x)<2(1-\Phi(x))$. Indeed, we can see that $\lim \limits_{x\rightarrow \infty} e^{tx} P(X>x)$ is less than twice that of a standard normal (which we already know to be light tailed, so we know that integral is finite even before we try to evaluate it).
I was somewhat surprised that you selected the linked skew-Cauchy (there are a number of others), because it's associated with a different skew-normal (which can be found within the article I linked to).
A somewhat similar approach to the one I used for the skew-normal can be used. Let's restrict consideration to the right half (i.e. consider only $x>\mu$). Then the density is proportional to the density of a Cauchy (because of the parameter $\lambda$), so $P(X>x)$ is proportional to that for a Cauchy. Consequently, the limit goes to infinity because the corresponding quantity for the Cauchy does.
So (without actually having to evaluate it in either case) that particular skew-normal is light tailed and that particular skew-Cauchy is heavy tailed.
