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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.

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