Janson 在1988年提出了反例，證明對於一個雙獨立隨機變數序列，中央極限定理是不成立的。我們進一步問：是否可以加上一些條件，使得中央極限定理成立？ 也就是說，我們想找出使雙獨立隨機變數序列具有中央極限定理的充分條件。
McLeish 在1974發表了一篇論文：”Dependent central limit theorem and invariance principles “，其中的定理2.1就提供了一個答案。這個定理有四個條件，其中兩個是針對所考慮的隨機變數序列本身，另外兩個是針對由此一隨機變數序列而定出的函數序列。但是這四個條件也不容易檢驗，我們希望可以找到比較容易檢驗而且可以推導出這四個條件的條件，那麼就可以取得實際運用上較大的便利。
本文假設其中兩個針對函數序列 (由所考慮的隨機變數序列而定出) 要求的條件成立，全力在另外兩個跟機率收斂有關的條件下工夫。先依Chandra所提出的Cesaro 均勻可積得到兩個定理；再依另一個由Hong提出，較Cesaro 均勻可積弱的條件得到其他的定理。
最後舉出一個雙獨立隨機變數序列作為例證。

hold. In my thesis, some related results are mentioned. I also give
some new version of conditions such that the central limit theorem would
hold for pairwise independent sequences. Finally, I give an example to
illustrate the results.

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