Generalizations of Some Concentration Inequalities

Abstract

For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $\varphi$, we first obtain a Chebyshev type inequality which provides an upper bound for $\varphi ({\lambda }_1)\mu (\{x\in \Omega :f(x)\ge {\lambda }_1\})+\sum _{k=2}^n\left(\varphi ({\lambda }_k)-\varphi ({\lambda }_{k-1})\right)\mu (\{x\in \Omega :f(x)\ge {\lambda }_k\}),$ where $0<{\lambda }_1<{\lambda }_2\cdots {\lambda }_n<\infty$. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaymé-Chebyshev, Cantelli and Hoeffding inequalities are obtained.

arXiv comment11 pages