Multivariate Chebyshev Inequality With Estimated Mean and Variance

Bartolomeo Stellato, Bart P.G. Van Parys, Paul J. Goulart

Research output: Contribution to journalArticlepeer-review

28 Scopus citations


A variant of the well-known Chebyshev inequality for scalar random variables can be formulated in the case where the mean and variance are estimated from samples. In this article, we present a generalization of this result to multiple dimensions where the only requirement is that the samples are independent and identically distributed. Furthermore, we show that as the number of samples tends to infinity our inequality converges to the theoretical multi-dimensional Chebyshev bound.

Original languageEnglish (US)
Pages (from-to)123-127
Number of pages5
JournalAmerican Statistician
Issue number2
StatePublished - Apr 3 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • General Mathematics
  • Statistics, Probability and Uncertainty


  • Chebyshev’s inequality
  • Probability bounds
  • Sampling


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