Two New Probability Inequalities with Limit Theorem Applications

Abstract

Two probability inequalities are established and each of them is applied to obtain a probability limit theorem. For a random variable $X$ and continuous nonnegative strictly increasing functions $G_{1}$ and $G_{2}$ defined on $[0,\infty)$ with $G_{i}(0)=0$, $\lim_{x\rightarrow\infty}G_{i}(x)=\infty$, $i=1,2$ and $G_{1}^{-1}(b_{n})=G_{2}^{-1}(b_{n})$, $n\geq 1$ where $\{b_{n},n\geq 1\}$ is a sequence of positive constants with $b_{n}\uparrow\infty$ as $n\rightarrow\infty$, the first inequality provides an upper bound for $\left|\mathbb{E}\left(G_{1}(|X|)\right)-\mathbb{E}\left(G_{2}(|X|)\right)\right|$ assuming that either $\mathbb{E}\left(G_{1}(|X|)\right)<\infty$ or $\mathbb{E}\left(G_{2}(|X|)\right)<\infty$. For a sequence of independent nonnegative random variables $\{X_{n},n\geq 1\}$ with partial sums $S_{n}=X_{1}+\cdots+X_{n}$, $n\geq 1$, the second inequality provides an upper bound for $\mathbb{P}\left(S_{n}>x+y\right)$ where $x$ and $y$ are nonnegative real numbers.