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Principle of conditioning in limit theorems for sums of random variables. (English) Zbl 0593.60031
Summary: Let $$\{X_{nk}:$$ $$k\in {\mathbb{N}}$$, $$n\in {\mathbb{N}}\}$$ be a double array of random variables adapted to the sequence of discrete filtrations $$\{$$ $$\{$$ $${\mathcal F}_{nk}:$$ $$k\in {\mathbb{N}}\cup \{0\}\}:$$ $$n\in {\mathbb{N}}\}$$. It is proved that for every weak limit theorem for sums of independent random variables there exists an analogous limit theorem which is valid for the system $$(\{X_{nk}\},\{{\mathcal F}_{nk}\})$$ and obtained by conditioning expectations with respect to the past. Functional extensions and connections with the martingale invariance principle are discussed.

##### MSC:
 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles
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