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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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