We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n-2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.
"Improved approximation of the sum of random vectors by the skew normal distribution." J. Appl. Probab. 51 (2) 466 - 482, June 2014. https://doi.org/10.1239/jap/1402578637