## The Annals of Probability

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

### 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

**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