Statistics and Probability Letters

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}) + n ∑ k=2 (φ(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.