Exact converses to a reverse AM–GM inequality, with applications to sums of independent random variables and (super)martingales
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For every given real value of the ratio μ:=A_X/G_X>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/G_X≥ v) and 𝖯(X/G_X≤ v) are obtained, in terms of μ and v. In particular, these bounds imply that X/G_X→1 in probability as A_X/G_X↓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) ≡ -x^2. 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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