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arXiv · 2108.01479

Generalizations of Some Concentration Inequalities

M. Ashraf Bhat·G. Sankara Raju Kosuru
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Abstract

For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $\phi$, we first obtain a Chebyshev type inequality which provides an upper bound for $\displaystyle \phi(\lambda_{1}) \mu(\{x \in \Omega : f(x) \geq \lambda_{1} \}) + \sum_{k=2}^{n}\left(\phi(\lambda_{k})- \phi(\lambda_{k-1})\right) \mu(\{x \in \Omega : f(x) \geq \lambda_{k}\}) ,$ where $0 < \lambda_1 < \lambda_2 \cdots \lambda_n < \infty$. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaym\'e-Chebyshev, Cantelli and Hoeffding inequalities are obtained.

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BibTeXRIS

M. Ashraf Bhat, G. Sankara Raju Kosuru. 2021-08-03. Generalizations of Some Concentration Inequalities. https://doi.org/10.1016/j.spl.2021.109298

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