What is a martingale array - its definition and importance?

Question

What is a martingale array? What is the importance of defining such an array, instead of using a martingale itself? A common example of this definition is a martingale difference array.

Answer 1

Define an array as a sequence of random variables on a probability space $(\Omega, \mathcal{F},P)$. Introduce doubly infinite arrays of random variables $X_{n,j},~\mathcal{F}_{n,j}$ for $j,n\geq 1$, and set sub $\sigma -$algebras of a $\sigma -$algebra. Adapting the array to the filtration, then $\{X_{n,j}\}$ is a MDA if

• the relation $\mathbb{E}X^2_{n,j}\equiv \sigma^2_{n,j}$ is finite $\forall n,j$

• we have the increasing embedding $\mathcal{F}_{n,j-1}\subset\mathcal{F}_{n,j}$

• We have zero expectation $\mathbb{E}_{j-1}(X_{n,j})=0 ~a.s,~\forall n,j$.

MDAs are used in limit issues in probability theory. An example of its use is that it can give generalisations of certain inequalities (such as the Chow-Birnbaum-Marshall submartingale maximal inequality) leading to a strong law of large numbers for martingale arrays with rows that are asymptotically stable.