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July, 1986 Principle of Conditioning in Limit Theorems for Sums of Random Variables
Adam Jakubowski
Ann. Probab. 14(3): 902-915 (July, 1986). DOI: 10.1214/aop/1176992446


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 $\{\{\mathscr{F}_{nk}: k \in \mathbb{N} \cup \{0\}\}: n \in \mathbb{N}\}$. 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 $(\{X_{nk}\}, \{\mathscr{F}_{nk}\})$ and obtained by conditioning expectations with respect to the past. Functional extensions and connections with the Martingale Invariance Principle are discussed.


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Adam Jakubowski. "Principle of Conditioning in Limit Theorems for Sums of Random Variables." Ann. Probab. 14 (3) 902 - 915, July, 1986.


Published: July, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0593.60031
MathSciNet: MR841592
Digital Object Identifier: 10.1214/aop/1176992446

Primary: 60F05
Secondary: 60F17

Keywords: martingale difference arrays , Martingale Invariance Principle , processes with independent increments , Random measures , tightness , Weak limit theorems for sums of random variables

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 3 • July, 1986
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