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On the Accuracy of the Exponential Approximation to Random Sums of Alternating Random Variables

Author

Listed:
  • Irina Shevtsova

    (Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
    Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia)

  • Mikhail Tselishchev

    (Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia)

Abstract

Using the generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with a finite non-zero first moment, we prove moment-type error-bounds in the Kantorovich distance for the exponential approximation to random sums of possibly dependent random variables with positive finite expectations, in particular, to geometric random sums, generalizing the previous results to alternating and dependent random summands. We also extend the notions of new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) distributions to alternating random variables in terms of the corresponding distribution functions and provide a criteria in terms of conditional expectations similar to the classical one. As corollary, we provide simplified error-bounds in the case of NBUE/NWUE conditional distributions of random summands.

Suggested Citation

  • Irina Shevtsova & Mikhail Tselishchev, 2020. "On the Accuracy of the Exponential Approximation to Random Sums of Alternating Random Variables," Mathematics, MDPI, vol. 8(11), pages 1-11, November.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:1917-:d:438602
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    Cited by:

    1. Alexander Bulinski & Nikolay Slepov, 2022. "Sharp Estimates for Proximity of Geometric and Related Sums Distributions to Limit Laws," Mathematics, MDPI, vol. 10(24), pages 1-37, December.

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