## The Annals of Probability

- Ann. Probab.
- Volume 19, Number 3 (1991), 1206-1226.

### Majorization, Exponential Inequalities and Almost Sure Behavior of Vector-Valued Random Variables

#### Abstract

In this paper we describe a general device that allows us to deduce various kinds of theorems (moment estimates, exponential inequalities, strong law of large numbers, stability results, bounded law of the iterated logarithm) for partial sums of independent vector-valued random variables from related results for partial sums of independent real-valued random variables. The concept of majorization will play a key role in our considerations.

#### Article information

**Source**

Ann. Probab., Volume 19, Number 3 (1991), 1206-1226.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990341

**Digital Object Identifier**

doi:10.1214/aop/1176990341

**Mathematical Reviews number (MathSciNet)**

MR1112413

**Zentralblatt MATH identifier**

0757.60002

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Secondary: 60F10: Large deviations 60F15: Strong theorems 60E15: Inequalities; stochastic orderings 60G50: Sums of independent random variables; random walks

**Keywords**

Majorization moment inequalities exponential inequalities strong law of large numbers bounded law of the iterated logarithm

#### Citation

Berger, Erich. Majorization, Exponential Inequalities and Almost Sure Behavior of Vector-Valued Random Variables. Ann. Probab. 19 (1991), no. 3, 1206--1226. doi:10.1214/aop/1176990341. https://projecteuclid.org/euclid.aop/1176990341