## The Annals of Mathematical Statistics

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

R. J. Tomkins

#### 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

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