Triangular array VS sequence of random variables

Question

To show types of convergence of random variable, I saw that people use triangular arrays instead of sequences of random variables. For example, for strictly stationary sequences $(Y_i,X_i)$ of random variables, one author shows that $\sup_{\lVert x\rVert\leq c_n}\lvert\hat\Psi(x)-E\hat\Psi(x)\rvert=O(a_n)$ where $\hat\Psi(x)=(nh^d)^{-1}\sum_{i=1}^n Y_i K((x-X_i)/h)$. Other author, without assume stationarity, use triagular arrays $(Y_{n,i},X_{n,i}), 1\leq i\leq n, n\geq 1$, to show the same result, but now for $\hat\Psi(x)=(nh^d)^{-1}\sum_{i=1}^n Y_{n,i} K((x-X_{n,i})/h)$.

My question

1. Is the results for triangular arrays imply the same results for sequences? My intuition say yes.
2. What is the advantage to use triangular arrays?

Answer 1

1. Yes, if the result is true for a triangular array, it is also true for a sequence. This is because every sequence can be turned into a triangular array. Consider the triangular array $T= \{T_1, T_2, \dots\}$ where

$$T_1 = \{X_1\}$$ $$T_2 = \{X_1, X_2\}$$ $$T_3 = \{X_1, X_2, X_3\}$$ $$\vdots$$ $$T_n = \{X_1, X_2, X_3, \dots, X_n\}$$ $$\vdots$$

Notice how $T_n$ is simply a sequence of numbers where the first $n-1$ values are just repeats of the previous array, so if you can say something about $T_n$ as part of a triangular array, you can certainly say the same about the sequence that equals $T_n$.

2. Using the previous example, the advantage here might now be obvious. If you are able to show a result for triangular arrays, you have a more general result then just one for sequences.