The Annals of Probability

Principle of Conditioning in Limit Theorems for Sums of Random Variables

Adam Jakubowski

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Abstract

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.

Article information

Source
Ann. Probab., Volume 14, Number 3 (1986), 902-915.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992446

Digital Object Identifier
doi:10.1214/aop/1176992446

Mathematical Reviews number (MathSciNet)
MR841592

Zentralblatt MATH identifier
0593.60031

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles

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

Citation

Jakubowski, Adam. Principle of Conditioning in Limit Theorems for Sums of Random Variables. Ann. Probab. 14 (1986), no. 3, 902--915. doi:10.1214/aop/1176992446. https://projecteuclid.org/euclid.aop/1176992446


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