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.