Exact converses to a reverse AM-GM inequality, with applications to sums of independent random variables and (super)martingales

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Department of Mathematical Sciences


For every given real value of the ratio μ := AX/GX> 1 of the arithmetic and geometric means of a positive random variable X and every real v > 0, exact upper bounds on the right- and left-tail probabilities P(X/GX≥ v) and P(X/GX→ v) are obtained, in terms of μ and v. In particular, these bounds imply that X/GX→ 1 in probability as AX/GX↓ 1. Such a result may be viewed as a converse to a reverse Jensen inequality for the strictly concave function f = ln, whereas the well-known Cantelli and Chebyshev inequalities may be viewed as converses to a reverse Jensen inequality for the strictly concave quadratic function f (x) Ξ -x2. As applications of the mentioned new results, improvements of the Markov, Bernstein-Chernoff, sub-Gaussian, and Bennett-Hoeffding probability inequalities are given.

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Mathematical Inequalities and Applications