Generalizations of Some Concentration Inequalities

TL;DR

This paper develops generalized concentration inequalities by extending classical results like Markov and Hoeffding inequalities through a new Chebyshev-type bound involving a nondecreasing function.

Contribution

It introduces a Chebyshev-type inequality that leads to generalized forms of several classical concentration inequalities.

Findings

Derived a Chebyshev-type inequality for measurable functions.

Generalized Markov, Chebyshev, Bienaymé-Chebyshev, Cantelli, and Hoeffding inequalities.

Provided bounds applicable to a wider class of functions and measures.

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.