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Delicate Comparison of the Central and Non-Central Lyapunov Ratios with Applications to the Berry–Esseen Inequality for Compound Poisson Distributions

Author

Listed:
  • Vladimir Makarenko

    (Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskie Gory, 1/52, 119991 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia)

  • Irina Shevtsova

    (Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskie Gory, 1/52, 119991 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia
    Federal Research Center “Informatics and Control”, Russian Academy of Sciences, Vavilov Str., 44/2, 119333 Moscow, Russia
    Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310005, China)

Abstract

For each t ∈ ( − 1 , 1 ) , the exact value of the least upper bound H ( t ) = sup { E | X | 3 / E | X − t | 3 } over all the non-degenerate distributions of the random variable X with a fixed normalized first-order moment E X 1 / E X 1 2 = t , and a finite third-order moment is obtained, yielding the exact value of the unconditional supremum M : = sup L 1 ( X ) / L 1 ( X − E X ) = 17 + 7 7 / 4 , where L 1 ( X ) = E | X | 3 / ( E X 2 ) 3 / 2 is the non-central Lyapunov ratio, and hence proving S. Shorgin’s (2001) conjecture on the exact value of M . As a corollary, an analog of the Berry–Esseen inequality for the Poisson random sums of independent identically distributed random variables X 1 , X 2 , … is proven in terms of the central Lyapunov ratio L 1 ( X 1 − E X 1 ) with the constant 0.3031 · H t ( 1 − t 2 ) 3 / 2 ∈ [ 0.3031 , 0.4517 ) , t ∈ [ 0 , 1 ) , which depends on the normalized first-moment t : = E X 1 / E X 1 2 of random summands and being arbitrarily close to 0.3031 for small values of t , an almost 1.5 size improvement from the previously known one.

Suggested Citation

  • Vladimir Makarenko & Irina Shevtsova, 2023. "Delicate Comparison of the Central and Non-Central Lyapunov Ratios with Applications to the Berry–Esseen Inequality for Compound Poisson Distributions," Mathematics, MDPI, vol. 11(3), pages 1-32, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:625-:d:1047410
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    References listed on IDEAS

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    1. von Chossy, R. & Rappl, G., 1983. "Some approximation methods for the distribution of random sums," Insurance: Mathematics and Economics, Elsevier, vol. 2(4), pages 251-270, October.
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