If we consider the above covariance as a matrix $$ C_{ij} =\mathrm{Cov}(X_i,\,X_j), $$ then the diagonal elements are just the familiar variances, \( C_{ii} = \mathrm{Cov}(X_i,\,X_i) = \mathrm{Var}(X_i) \). It turns out that all the off-diagonal elements are zero if the stochastic variables are uncorrelated.