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Improvements of the Markov and Chebyshev inequalities using the partial expectation

Abstract The multivariate Markov and multiple Chebyshev inequalities are improved using the partial expectation. The conditions of the strict inequalities between the improved and unimproved bounds of probabilities are given. The known upper bounds of the tail probabilities by the partial moment are derived as special cases of the Markov inequality, whose improved bounds and extensions to multivariate cases using the semicovariances and partial product moment are presented.