It is well known that the variance of the sum of independent random variables (not necessarily i.i.d.) is the sum of the variance of each random variable (i.e. $Var[X\_1 + X\_2 ... X\_n] = \sum\_{i=1}^{n} Var[X\_i]$). What about the higher absolute moment? For instance, does the following equation hold?
\begin{equation}
E\left[\left| \sum\_{i=1}^n X\_i - E\left[\sum\_{i=1}^n X\_i\right] \right|^3\right] = \sum\_{i=1}^n E\left[\left| X\_i - E\left[X\_i\right] \right|^3\right]. 
\end{equation}



High absolute moment of sum of the independent random variables
The generalization to higher powers of the additivity statement of the variance goes via the cumulants: If two variables $X$ and $Y$ are independent, then the cumulants $\kappa\_n$ are additive, $\kappa\_n(X+Y)=\kappa\_n(X)+\kappa\_n(Y)$.


It follows from this additivity that third central moments are additive, $$E\left[\left(\sum\_i X\_i-\sum\_i E[X\_i]\right)^3\right]=\sum\_i E\left[\left(X\_i-E[X\_i]\right)^3\right],$$
iff the variables $X\_i$ all have the same variance. No such simple additivity criterion exists for absolute values of third powers.


