Markov–Krein theorem explained

In probability theory, the Markov–Krein theorem gives the best upper and lower bounds on the expected values of certain functions of a random variable where only the first moments of the random variable are known.^{[1] [2] [3] [4]} The result is named after Andrey Markov and Mark Krein.^[5]

The theorem can be used to bound average response times in the M/G/k queueing system.^[6]

Notes and References

1. Stokes . S. Lynne . S. Lynne Stokes . Mary H.. Mulry-Liggan . Mary Mulry . Estimation of Interviewer Variance for Categorical Variables . Journal of Official Statistics . 3 . 1987 . 389–401 . 11 June 2013.
2. Brockett . P. L. . Kahane . Y. . 10.1287/mnsc.38.6.851 . Risk, Return, Skewness and Preference . Management Science . 38 . 6 . 851 . 1992 .
3. Simar . L.. Maximum Likelihood Estimation of a Compound Poisson Process . 10.1214/aos/1176343651 . The Annals of Statistics. 4 . 6 . 1200 . 1976 . 2958588. free .
4. Book: S.. Karlin . Samuel Karlin . W. J. . Studden . Tchebycheff Systems, with Applications in Analysis and Statistics . Interscience . New York . 1966 . 82.
5. Kreĭn . M. G.. Mark Krein. The ideas of P. L. Čebyšev and A. A. Markov in the theory of limiting values of integrals and their further development . Amer. Math. Soc. Transl. . 2 . 12. 1959 . 1–121. 0113106 .
6. Gupta . V. . Osogami . T. . 10.1007/s11134-011-9248-8 . On Markov–Krein characterization of the mean waiting time in M/G/K and other queueing systems . Queueing Systems . 68 . 3–4 . 339 . 2011 .