Generalizations of some concentration inequalities

Abstract

For a real-valued measurable function f and a nonnegative, nondecreasing function ϕ, we first obtain a Chebyshev type inequality which provides an upper bound for ϕ(λ1)μ({x∈Ω:f(x)≥λ1})+∑k=2nϕ(λk)−ϕ(λk−1)μ({x∈Ω:f(x)≥λk}), where 0<λ1<λ2<⋯<λn<∞. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaymé–Chebyshev, Cantelli and Hoeffding inequalities are obtained.