While studying Stable Law we understand that when alfa (tail index parameter) is 2, then regardless of beta (skewness parameter) the random variable is normal. This may be viewed by checking the Characteristic function of the stable distributed random variable. But my question is different, What if I would like to view the skewed normal distribution as a part that belongs to the stable law? Is this wrong to make a relation between this and the stable distribution. Particularly when we think about the shape parameter in the Skewed normal distribution as the skewness parameter in the stable law?
Thank you very much in advance
The skew normal distribution has finite variance for all $\alpha$. All stable laws with finite variance are Gaussian (without skew). So, if the skewness parameter $\alpha\neq0$, the skew normal is not stable. To put it differently, the skew normal is either skewed or stable, but not both.