In this article( http://www.diva-portal.org/smash/get/diva2:302313/FULLTEXT01.pdf )page 28 explains how to derive the skewness of a sum of random variables; I haven't been able to derive this expression in case of dealing with a difference of random variables. The final dexpression for the skewness of the sum of independent random variables is:
skewness(X+Y)=(μ3(X)+μ3(Y))/(μ2(X)+μ2(Y))^3/2
Skewness is defined by:
$$\text{skewness}(Z) = \frac{\mu _3}{\mu _2^{3/2}}$$
where $\mu_k$ denotes the $k^{th}$ central moment of $Z$.
If $Z = X + Y$ and given independence, the formula becomes:
$$\text{skewness}(X+Y) = \frac{\mu _3(X) + \mu _3(Y)}{\big(\mu _2(X) + \mu _2(Y)\big)^{3/2}}$$
If $Z = X - Y$ and given independence, the formula becomes:
$$\text{skewness}(X-Y) = \frac{\mu _3(X) - \mu _3(Y)}{\big(\mu _2(X) + \mu _2(Y)\big)^{3/2}}$$
The denominator does not change since $\text{Var}(X-Y) = \text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y)$
