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

\begin{aligned} X_{11} \\ X_{21} X_{22} \\ X_{31} X_{32} X_{33} \\ \dots, \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 $X_{n i} = Y_{n i} - μ_{n i}$
I’ve stated the definition here in terms of scalar variables, but the entries in this triangle can also be random vectors $x_{n i}$

Row-wise sums

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

s_{n}^{2} = V Z_{n} = \sum_{i = 1}^{n} V X_{n i} = \sum_{i = 1}^{n} σ_{n i}^{2}

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

V_{n} = V z_{n} = \sum_{i = 1}^{n} V x_{n i} = \sum_{i = 1}^{n} Σ_{n i} .