The echelon Markov and Chebyshev inequalities

Abstract

A multivariate version of the sharp Markov inequality is derived, when associated probabilities are extended to segments of the supports of non-negative random variables, where the probabilities take echelon forms. It is shown that when some positive lower bounds of these probabilities are available, the multivariate Markov inequality without the echelon forms is improved. The corresponding results for Chebyshev’s inequality are also obtained.

Keywords:

• ABC Theorem
• expectation
• monotonic transformation
• probability inequality
• sharpness
• survival function

Additional information

Funding

This work was partially supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology (JSPS KAKENHI, Grant No. 17K00042).