## Annals of Probability

### Principle of Conditioning in Limit Theorems for Sums of Random Variables

#### 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

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