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.
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Funding
The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (grant no. RGPIN-2014-05428).
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(Submitted by A. I. Volodin)
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Li, D., Rosalsky, A. Two New Probability Inequalities with Limit Theorem Applications. Lobachevskii J Math 42, 336–341 (2021). https://doi.org/10.1134/S1995080221020141
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DOI: https://doi.org/10.1134/S1995080221020141