On the infinite divisibility of some skewed symmetric distributions

Infinite divisibility of some symmetric distributions skewed by an additive component is investigated. We find in particular that the skew-normal distribution of Azzalini [1985. A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178] and the multivariate skew-normal distribution of Azzalini and Dalla Valle [1996. The multivariate skew-normal distribution. Biometrika 83, 715–726] are not infinitely divisible.