0
$\begingroup$

I was wondering if anyone out there would know of an appropriate central limit theorem (or be able to apply a central limit theorem type argument) that would allow me to find an asymptotic distribution/weak convergence of the following Martingale Difference Sequence (MDS):

Let $\left\{\xi_j^n\right\}_{j=1}^n \sim $ MDS $(0,\sigma^4)$. Then:

\begin{equation*} \frac{1}{\sqrt{n}}\sum_{j=1}^n \xi_j^n \quad \overset{\mathcal{D}}\longrightarrow \quad ?? \end{equation*}

By the ....??.... theorem under ...?... (some assumptions), as $n\rightarrow\infty$?

I now this is not a very "well posed" problem per se, and if there are any other "assumptions" that are required/needed in order to obtain this sort of weak convergence please let me know. Also could you also please provide a reference/references so that I may study this in a little bit more depth myself.

Many thanks

  • 0
    Sorry forgot to also mention, is there also a extension to this much like the function central limit theorem so that we may obtain an asymptotic distribution/weak convergence of the following expression: \begin{equation*}\frac{1}{\sqrt{n}} \sum_{j=1}^{\lfloor{nr}\rfloor} \xi_j^n\end{equation*} – 2017-01-11

1 Answers 1

0

You could use the result of McLeish, which is perfectly suited to the setting of arrays of martingale differences.