How come does the variance sum law work for more than 2 independent random variables?

Question

I have learnt through proof that the variance sum law indeed works for two independent random variables, such as var(x + y) = var(x) + var(y). But what I can’t wrap my head around is how this is possible for more than two independent random variables (i.e. x, y, and z). Covariance itself measures the relationship between variance in one variable to variance in another, and when adding more than 2 independent random variables, it cannot simply be reduced to 0. I am asking this question, because I have seen too many proofs of the variance of the sample mean that involve using the variance sum law, and in all cases, those proofs simply apply the variance sum law no matter how many random variables they are adding. I would highly appreciate an intuitive, mind clearing insight.

Answer 1

We have

\begin{align} Var\left( \sum_{i=1}^n X_i\right) &= Cov\left( \sum_{i=1}^n X_i, \sum_{i=1}^n X_i\right)\\ &=\sum_{i=1}^n \sum_{j=1}^n Cov(X_i, X_j) \\ &= \sum_{i=1}^n Var(X_i) + 2\sum_{i < j}Cov(X_i, X_j) \end{align}

Hence if the pairwise covariance is $0$, we have sum of variance is equal to variance of the sum.