East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

ON RECURSIONS FOR MOMENTS OF A COMPOUND RANDOM VARIABLE: AN APPROACH USING AN AUXILIARY COUNTING RANDOM VARIABLE

  • Yoora Kim (Department of Mathematics, University of Ulsan)
  • Received : 2023.03.28
  • Accepted : 2023.05.25
  • Published : 2023.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

We present an identity on moments of a compound random variable by using an auxiliary counting random variable. Based on this identity, we develop a new recurrence formula for obtaining the raw and central moments of any order for a given compound random variable.

Keywords

Acknowledgement

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2019R1F1A1060743).

References

  1. N. De Pril, Moments of a class of compound distributions, Scandinavian Actuarial Journal, 1986(2) (1986), 117-120. https://doi.org/10.1080/03461238.1986.10413800
  2. R.W. Grubbstrom and O. Tang, The moments and central moments of a compound distribution, European Journal of Operational Research, 170(1) (2006), 106-119. https://doi.org/10.1016/j.ejor.2004.06.012
  3. O. Hesselager, A recursive procedure for calculation of some compound distributions, ASTIN Bulletin: The Journal of the IAA, 24(1) (1994), 19-32. https://doi.org/10.2143/AST.24.1.2005078
  4. D. Kim and Y. Kim, Recurrence relations for higher order moments of a compound binomial random variable, East Asian Mathematical Journal, 34(1) (2018), 59-67. https://doi.org/10.7858/EAMJ.2018.007
  5. T. Loc Hung, On the rates of convergence in central limit theorems for compound random sums of independent random variables, Lobachevskii Journal of Mathematics, 42(2) (2021), 374-393. https://doi.org/10.1134/S1995080221020128
  6. T. Loc Hung, On the weak laws of large numbers for compound random sums of independent random variables with convergence rates, Bulletin of the Iranian Mathematical Society, 48(4) (2022), 1967-1989. https://doi.org/10.1007/s41980-021-00632-5
  7. M. Murat, Recurrence relations for moments of doubly compound distributions, International Journal of Pure and Applied Mathematics, 79(3) (2012), 481-492.
  8. M. Murat and D. Szynal, On moments of counting distributions satisfying the kth-order recursion and their compound distributions, Journal of Mathematical Sciences, 92(4) (1998), 4038-4043. https://doi.org/10.1007/BF02432340
  9. M. Murat and D. Szynal, On computational formulas for densities and moments of compound distributions, Journal of Mathematical Sciences, 99(3) (2000), 1286-1299. https://doi.org/10.1007/BF02674088
  10. H.H. Panjer, Recursive evaluation of a family of compound distributions, ASTIN Bulletin: The Journal of the IAA, 12(1) (1981), 22-26. https://doi.org/10.1017/S0515036100006796
  11. E. Pekoz and S.M. Ross, Compound random variables, Probability in the Engineering and Informational Sciences, 18(4) (2004), 473-484. https://doi.org/10.1017/S0269964804184039
  12. K.J. Schroter, On a family of counting distributions and recursions for related compound distributions, Scandinavian Actuarial Journal, 1990(2-3) (1990), 161-175. https://doi.org/10.1080/03461238.1990.10413879
  13. S.K. Seong, Recurrence formulas for the raw moments and the central moments of compound random variables, Master's Thesis, University of Ulsan, 2020 (2020), 1-20.
  14. B. Sundt, On some extensions of Panjer's class of counting distributions, ASTIN Bulletin: The Journal of the IAA, 22(1) (1992), 61-80. https://doi.org/10.2143/AST.22.1.2005127
  15. B. Sundt, Some recursions for moments of compound distributions, Insurance: Mathematics and Economics, 33(3) (2003), 487-496. https://doi.org/10.1016/j.insmatheco.2003.09.002
  16. F. Tank and S. Eryilmaz, On bivariate compound sums, Journal of Computational and Applied Mathematics, 365(1) (2020), 112371.