Row-wise independent centered triangular array

For each integer $n\ge 1$, let $k_n\ge 1$ be a finite integer and let $X_{n,1},\dots ,X_{n,k_n}$ be integrable real random variables on a probability space $({\mathrm{\Omega }}_n,{\mathcal{F}}_n,{\mathbb{P}}_n)$. The family is a centered triangular array when ${\mathbb{E}}_n{X}_{n,k}=0$ for every admissible pair $(n,k)$. It is row-wise independent when, for each fixed n and all Borel sets $B_1,\dots ,B_{k_n}$, ${\mathbb{P}}_n\text{\hspace{0.17em}⁣}({\bigcap }_{k=1}^{k_n}\{X_{n,k}\in B_k\})={\prod }_{k=1}^{k_n}{\mathbb{P}}_n(X_{n,k}\in B_k)\mathrm{.}$ Taking unused $B_k$ equal to the real line gives the same factorization for each subfamily; by Independent random elements are characterized by finite rectangle probabilities, this is exactly independence in Independent random elements. Random variables and integrability have the meanings in Random elements and real random variables and Expectation of a nonnegative or integrable random variable. There is no independence requirement between different rows. The spaces may differ with n; assertions about row sums compare their laws. A single-entry row is independent automatically, and deterministic zero entries are allowed. Empty rows are excluded by $k_n\ge 1$. This definition makes no existence or choice assumption.