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.

中文翻译：

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极限定理应用的两个新的概率不等式

摘要

建立两个概率不等式，并应用每个概率不等式以获得概率极限定理。对于随机变量\（X \）和连续非负严格递增函数\（G_ {1} \）和\（G_ {2} \）在\（[0，\ infty）\）上使用\（G_ {i }（0）= 0 \），\（\ lim_ {x \ rightarrow \ infty} G_ {i}（x）= \ infty \），\（i = 1,2 \）和\（G_ {1} ^ {-1}（b_ {n}）= G_ {2} ^ {-1}（b_ {n}）\），\（n \ geq 1 \）其中\（\ {b_ {n}，n \ geq 1 \} \）是一个正常数序列，其中\（b_ {n} \ uparrow \ infty \）为\（n \ rightarrow \ infty \），第一个不等式为\（\ left | \ mathbb {E} \ left（G_ {1}（| X |）\ right）-\ mathbb {E} \ left（G_ {2}（| X |）\ right）\ right | \）假设\（\ mathbb {E} \ left（G_ {1}（| X |）\ right）<\ infty \）或\（\ mathbb {E} \ left （G_ {2}（| X |）\ right）<\ infty \）。对于具有部分和\（S_ {n} = X_ {1} + \ cdots + X_ {n} \）的独立非负随机变量\（\ {X_ {n}，n \ geq 1 \} \）的序列，\（n \ geq 1 \），第二个不等式为\（\ mathbb {P} \ left（S_ {n}> x + y \ right）\）提供一个上限，其中\（x \）和\（y \）是非负实数。