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.
Publié le : 1986-07-14
Classification:
Weak limit theorems for sums of random variables,
martingale difference arrays,
Martingale Invariance Principle,
processes with independent increments,
random measures,
tightness,
60F05,
60F17
@article{1176992446,
author = {Jakubowski, Adam},
title = {Principle of Conditioning in Limit Theorems for Sums of Random Variables},
journal = {Ann. Probab.},
volume = {14},
number = {4},
year = {1986},
pages = { 902-915},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992446}
}
Jakubowski, Adam. Principle of Conditioning in Limit Theorems for Sums of Random Variables. Ann. Probab., Tome 14 (1986) no. 4, pp. 902-915. http://gdmltest.u-ga.fr/item/1176992446/