대한수학회지 (Journal of the Korean Mathematical Society)

  • 제61권5호
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  • Pages.853-874
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  • 2024
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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LARGE DEVIATIONS FOR A SUPER-HEAVY TAILED 𝛽-MIXING SEQUENCE

  • Yu Miao (College of Mathematics and Information Science Henan Normal University) ;
  • Qing Yin (College of Mathematics and Information Science Henan Normal University)
  • 투고 : 2022.11.18
  • 심사 : 2024.04.25
  • 발행 : 2024.09.01
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⟨ 이전 논문 다음 논문 ⟩

초록

Let {X, Xn; n ≥ 1} be a 𝛽-mixing sequence of identical nonnegative random variables with super-heavy tailed distributions and Sn = X1 + X2 + · · · + Xn. For 𝜀 > 0, b > 1 and appropriate values of x, we obtain the logarithmic asymptotics behaviors for the tail probabilities ℙ(Sn > e𝜀nx) and P(Sn > e𝜀bn). Moreover, our results are applied to the log-Pareto distribution and the distribution for the super-Petersburg game.

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과제정보

The authors would like to express their sincere gratitude to the anonymous referees and editor for their helpful comments which led to an improved presentation of this paper.

참고문헌

  1. K. B. Athreya and S. G. Pantula, Mixing properties of Harris chains and autoregressive processes, J. Appl. Probab. 23 (1986), no. 4, 880-892. https://doi.org/10.1017/s0021900200118674
  2. H. Berbee, Convergence rates in the strong law for bounded mixing sequences, Probab. Theory Related Fields 74 (1987), no. 2, 255-270. https://doi.org/10.1007/BF00569992
  3. R. C. Bradley, Basic properties of strong mixing conditions, Dependence in Probability and Statistics (Oberwolfach, 1985), 165-192, Progr. Probab. Statist. 11, Birkhauser Boston, Boston, MA, 1986. https://doi.org/10.1007/978-1-4615-8162-8_8
  4. X. Chen, Q. Shao, W. B. Wu, and L. Xu, Self-normalized Cramer-type moderate deviations under dependence, Ann. Statist. 44 (2016), no. 4, 1593-1617. https://doi.org/10.1214/15-AOS1429
  5. P. Embrechts, C. Kluppelberg, and T. Mikosch, Modelling Extremal Events, Applications of Mathematics (New York), 33, Springer, Berlin, 1997. https://doi.org/10.1007/978-3-642-33483-2
  6. M. Falk, J. Husler, and R.-D. Reiss, Laws of Small Numbers: Extremes and Rare Events, third, revised and extended edition, Birkhauser/Springer Basel AG, Basel, 2011. https://doi.org/10.1007/978-3-0348-0009-9
  7. J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 1978.
  8. N. Gantert, A note on logarithmic tail asymptotics and mixing, Statist. Probab. Lett. 49 (2000), no. 2, 113-118. https://doi.org/10.1016/S0167-7152(00)00037-7
  9. L. Gao, Q.-M. Shao, and J. Shi, Refined Cramer-type moderate deviation theorems for general self-normalized sums with applications to dependent random variables and winsorized mean, Ann. Statist. 50 (2022), no. 2, 673-697. https://doi.org/10.1214/21-aos2122
  10. Y. Hu and H. Nyrhinen, Large deviations view points for heavy-tailed random walks, J. Theoret. Probab. 17 (2004), no. 3, 761-768. https://doi.org/10.1023/B:JOTP.0000040298.43712.e8
  11. D. Li and Y. Miao, A supplement to the laws of large numbers and the large deviations, Stochastics 93 (2021), no. 8, 1261-1280. https://doi.org/10.1080/17442508.2021.1903465
  12. D. Li, Y. Miao, and G. Stoica, A general large deviation result for partial sums of i.i.d. super-heavy tailed random variables, Statist. Probab. Lett. 184 (2022), Paper No. 109371, 10 pp. https://doi.org/10.1016/j.spl.2022.109371
  13. Y. Liu and Y. J. Hu, Large deviations viewpoints for a heavy-tailed β-mixing sequence, Sci. China Ser. A 48 (2005), no. 11, 1554-1566. https://doi.org/10.1360/03ys0289
  14. H. Masuda, Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps, Stochastic Process. Appl. 117 (2007), no. 1, 35-56. https://doi.org/10.1016/j.spa.2006.04.010
  15. Y. Miao and D. Li, A general logarithmic asymptotic behavior for partial sums of i.i.d. random variables, Statist. Probab. Lett. 208 (2024), Paper No. 110043, 11 pp. https://doi.org/10.1016/j.spl.2024.110043
  16. Y. Miao, Y. Wang, and X. Ma, Some limit results for Pareto random variables, Comm. Statist. Theory Methods 42 (2013), no. 24, 4384-4391. https://doi.org/10.1080/03610926.2011.650271
  17. Y. Miao, T. Xue, K. Wang, and F. Zhao, Large deviations for dependent heavy tailed random variables, J. Korean Statist. Soc. 41 (2012), no. 2, 235-245. https://doi.org/10.1016/j.jkss.2011.09.001
  18. Y. Miao and Q. Yin, Limit behaviors for a heavy-tailed β-mixing random sequence, Lith. Math. J. 63 (2023), no. 1, 92-103. https://doi.org/10.1007/s10986-022-09584-7
  19. T. Nakata, Limit theorems for super-heavy tailed random variables with truncation: application to the super-Petersburg game, Bull. Inst. Math. Acad. Sin. (N.S.) 15 (2020), no. 2, 123-141. https://doi.org/10.21915/BIMAS.2020202
  20. T. Nakata, Large deviations for super-heavy tailed random walks, Statist. Probab. Lett. 180 (2022), Paper No. 109240. https://doi.org/10.1016/j.spl.2021.109240
  21. T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels, Stochastic Processes for Insurance and Finance, Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1999. https://doi.org/10.1002/9780470317044
  22. G. Schwarz, Finitely determined processes-an indiscrete approach, J. Math. Anal. Appl. 76 (1980), no. 1, 146-158. https://doi.org/10.1016/0022-247X(80)90068-2
  23. G. Stoica, Large gains in the St. Petersburg game, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 563-566. https://doi.org/10.1016/j.crma.2008.03.026