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}
1 Answer
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, $$E\left[\left(\sum_i X_i\sum_i E[X_i]\right)^3\right]=\sum_i E\left[\left(X_iE[X_i]\right)^3\right],$$ iff the variables $X_i$ all have the same variance. No such simple additivity criterion exists for absolute values of third powers.

$\begingroup$ Thank you for the clarification! $\endgroup$ Commented May 27, 2021 at 11:11