## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 43, Number 1 (1972), 230-235.

### Some Probability Inequalities Related to the Law of Large Numbers

#### Abstract

Let $S_1, S_2,\cdots, S_n$ be integrable random variables (rv). Upper bounds of the Hajek-Renyi type are presented for $P(\max_{1\leqq k\leqq n} \phi_k S_k \geqq \varepsilon \mid \mathscr{G})$ where $\phi_1 \geqq \cdots \geqq \phi_n > 0$ are rv, $\varepsilon > 0$ and $\mathscr{G}$ is a $\sigma$-field. The theorems place no further assumptions on the $S_k$'s; some, in fact, do not even require the integrability. It is shown, however, that if the $S_k$'s are partial sums of independent rv or if $S_1, S_2,\cdots, S_n$ forms a submartingale, then some well-known inequalities follow as consequences of these theorems.

#### Article information

**Source**

Ann. Math. Statist., Volume 43, Number 1 (1972), 230-235.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177692715

**Digital Object Identifier**

doi:10.1214/aoms/1177692715

**Mathematical Reviews number (MathSciNet)**

MR298740

**Zentralblatt MATH identifier**

0238.60023

**JSTOR**

links.jstor.org

#### Citation

Tomkins, R. J. Some Probability Inequalities Related to the Law of Large Numbers. Ann. Math. Statist. 43 (1972), no. 1, 230--235. doi:10.1214/aoms/1177692715. https://projecteuclid.org/euclid.aoms/1177692715