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.
1 Answer
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.
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$\begingroup$ Thanks.Could you give a general definition of arrays? and what do you mean "rows" what forms rows? $\endgroup$ Jul 23, 2015 at 16:08
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$\begingroup$ The strong law of large numbers states that a given sample average converges almost surely to the expectation. We can achieve a strong LLN for MDAs such that the rows of the arrays are 'well-behaved' or 'nice' in an asymptotic sense. I will add a definition of the arrays. $\endgroup$– user230715Jul 23, 2015 at 16:15
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$\begingroup$ "Define an array as a sequence of random variables" What? Actually the whole point of considering arrays is to deal with doubly indexed families of random variables. $\endgroup$– DidJul 23, 2015 at 22:49