The Annals of Mathematical Statistics

Some Probability Inequalities Related to the Law of Large Numbers

R. J. Tomkins

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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


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