Definition: A triangular array of random variables is of the form

\[\begin{aligned} &X_{11} \\ &X_{21} \quad X_{22} \\ &X_{31} \quad X_{32} \quad X_{33} \\ &\ldots, \end{aligned}\]

where the random variables in each row (i) are independent of each other, (ii) have zero mean and (iii) have finite variance.

Note that:

The requirement that the variables have zero mean is only for convenience; we can always construct zero-mean variables by considering \(X_{ni} = Y_{ni}-\mu_{ni}\)
I’ve stated the definition here in terms of scalar variables, but the entries in this triangle can also be random vectors \(\x_{ni}\)

Row-wise sums

Let \(Z_n = \sum_{i=1}^n X_{ni}\) denote the row-wise sum of the array. Since the elements of each row are independent, we have

\[s_n^2 = \Var Z_n = \sum_{i=1}^n \Var X_{ni} = \sum_{i=1}^n \sigma_{ni}^2\]

or, if the elements in the array are random vectors,

\[\V_n = \Var \z_n = \sum_{i=1}^n \Var \x_{ni} = \sum_{i=1}^n \bS_{ni}.\]