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Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing, Maximum, and Minimum

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Closure Properties for Heavy-Tailed and Related Distributions

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Abstract

We start this chapter with a motivating discussion on the use of convolution closure when evaluating the ruin probability in the classical risk process. In Sect. 3.3, we discuss the convolution closure properties in relation to the notion of max-sum equivalence. In further sections, we overview and discuss the closure properties of the heavy-tailed and related distributions, introduced in Chap. 2, under strong/weak tail-equivalence, convolution, finite mixing, maximum, and minimum. Together, we show how these closure properties can be extended to the convolution power and order statistics. The corresponding closure properties are followed by discussions, numerous examples, and counterexamples.

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References

  1. Albrecher, H., Asmussen, S., Kortschak, D.: Tail asymptotics for the sum of two heavy-tailed dependent risks. Extremes 9, 107–130 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aleškevičienė, A., Leipus, R., Šiaulys, J.: Second-order asymptotics of ruin probabilities for semiexponential claims. Lith. Math. J. 49, 364-371 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Asmussen, S., Albrecher, H.: Ruin Probabilities, 2nd edn. World Scientific, Singapore (2010)

    Book  MATH  Google Scholar 

  4. Baltrūnas, A.: Second order behaviour of ruin probabilities. Scand. Actuar. J. 2, 120–133 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baltrūnas, A.: On the subexponentiality property of a class of random variables. Math. Notes 69, 571–574 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baltrūnas, A.: Second order behaviour of ruin probabilities in the case of large claims. Insurance Math. Econom. 36, 485–498 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baltrūnas, A., Omey, E., Van Gulck, S.: Hazard rates and subexponential distributions. Publications de l’Institut Mathématique 80, 29–46 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beck, S., Blath, J., Scheutzow, M.: A new class of large claim size distributions: definition, properties, and ruin theory. Bernoulli 21, 2475–2483 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Berkes, I., Györfi, L., Kevei, P.: Tail probabilities of St. Petersburg sums, trimmed sums, and their limit. J. Theor. Probab. 30, 1104–1129 (2017)

    Google Scholar 

  10. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)

    Book  MATH  Google Scholar 

  11. Cai, J., Tang, Q.: On max-type equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Probab. 41, 117–130 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cheng, D., Ni, S., Pakes, A.G., Wang, Y.: Some properties of the exponential distribution class with applications to risk theory. J. Korean Stat. Soc. 41, 515–527 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cheng, D., Wang, Y.: Asymptotic behavior of the ratio of tail probabilities of sum and maximum of independent random variables. Lith. Math. J. 52, 29–39 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chover, J., Ney, P., Wainger, S.: Degeneracy properties of subcritical branching processes. Ann. Probab. 1, 663–673 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cline, D.B.H.: Convolutions tails, product tails and domains of attraction. Probab. Theory Relat. Fields 72, 529–557 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cui, Z., Omey, E., Wang, W., Wang, Y.: Asymptotics of convolution with the semi-regular-variation tail and its application to risk. Extremes 21, 509–532 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cui, Z., Wang, Y.: On the long tail property of product convolution. Lith. Math. J. 60, 315–329 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Danilenko, S., Šiaulys, J., Stepanauskas, G.: Closure properties of O-exponential distributions. Statist. Probab. Lett. 140, 63–70 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Denisov, D., Foss, S., Korshunov, D.: Tail asymptotics for the supremum of random walk when the mean is not finite. Queueing Syst. 46, 15–33 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Denisov, D., Foss, S., Korshunov, D.: On lower limits and equivalences for distribution tails of randomly stopped sums. Bernoulli 14, 391–404 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dindienė, L., Leipus, R.: Weak max-sum equivalence for dependent heavy-tailed random variables. Lith. Math. J. 56, 49–59 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dirma, M., Paukštys, S., Šiaulys J.: Tails of the moments for sums with dominatedly varying random summands. Mathematics 9(8), 824 (2021)

    Article  Google Scholar 

  23. Embrechts, P., Goldie, C.M.: On closure and factorization properties of subexponential and related distributions. J. Aust. Math. Soc. Ser. A 29, 243–256 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Embrechts, P., Goldie, C.M.: On convolution tails. Stochastic Processes Appl. 13, 263–278 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  25. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, New York (1997)

    Book  MATH  Google Scholar 

  26. Embrechts, P., Veraverbeke, N.: Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  27. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1971)

    MATH  Google Scholar 

  28. Finkelshtein, D., Tkachov, P. Kesten’s bound for subexponential densities on the real line and its multi-dimensional analogues. Adv. Appl. Probab. 50, 373–395 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Foss, S., Korshunov, D.: Lower limits and equivalences for convolution tails. Ann. Probab. 35, 366–383 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Foss, S., Korshunov, D., Zachary, S.: Convolutions of long-tailed and subexponential distributions. J. Appl. Probab. 46, 756–767 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Foss, S., Korshunov, D., Zachary, S.: An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd edn. Springer, New York (2013)

    Book  MATH  Google Scholar 

  32. Geluk, J.L.: Some closure properties for subexponential distributions. Statist. Probab. Lett. 79, 1108–1111 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Geluk, J., de Haan, L., Resnick, S., Stărică, C.: Second-order regular variation, convolution and the central limit theorem. Stochastic Processes Appl. 69, 139–159 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Geluk, J.L., Frenk, J.B.G.: Renewal theory for random variables with a heavy tailed distribution and infinite variance. Statist. Probab. Lett. 81, 77–82 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Geluk, J., Tang, Q.: Asymptotic tail probabilities of sums of dependent subexponential random variables. J. Theor. Probab. 22, 871–882 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gerber, H.U.: An introduction to mathematical risk theory. In: Hübner Foundation Monograph, vol. 8. Irwin, Homewood (1979)

    Google Scholar 

  37. Grandell, J.: Aspects of Risk Theory. Springer, New York (1991)

    Book  MATH  Google Scholar 

  38. Huang, Y., Li, Y., Xu, M.: Analysis of order statistics from distributions with regularly varying tails. Commun. Stat.- Theory Methods 43, 2702–2713 (2014)

    Google Scholar 

  39. Jiang, T., Wang, Y., Cui, Z., Chen, Y.: On the almost decrease of a subexponential density. Statist. Probab. Lett. 153, 71–79 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kizinevič, E., Sprindys, J., Šiaulys, J.: Randomly stopped sums with consistently varying distributions. Modern Stochastics: Theory and Applications 3, 165–179 (2016)

    MathSciNet  MATH  Google Scholar 

  41. Klüppelberg, C.: Subexponential distributions and integrated tails. J. Appl. Probab. 25, 132–141 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  42. Klüppelberg, C.: Asymptotic ordering of distribution functions and convolution semigroups. Semigroup Forum 40, 77–92 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  43. Klüppelberg, C.: Asymptotic ordering of risks and ruin probabilities. Insurance Math. Econom. 12, 259–264 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  44. Klüppelberg, C., Villasenor, J.A.: The full solution of the convolution closure problem for convolution-equivalent distributions. J. Math. Anal. Appl. 160, 79–92 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  45. Konstantinides, D.G.: Risk Theory: A Heavy Tail Approach. World Scientific, New Jersey (2018)

    MATH  Google Scholar 

  46. Konstantinides, D.G., Leipus, R., Šiaulys, J.: On the convolution closure problem for strong subexponential distributions. Nonlinear Anal. Modell. Control 28, 97–115 (2023)

    MATH  Google Scholar 

  47. Leipus, R., Šiaulys, J., Konstantinides, D.: Minimum of heavy-tailed random variables is not heavy-tailed. AIMS Math. 8, 13066–13072 (2023)

    Article  MathSciNet  Google Scholar 

  48. Leipus, R., Surgailis, D.: On long-range dependence in regenerative processes based on a general ON/OFF scheme. J. Appl. Probab. 44, 379–392 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  49. Leipus, R., Šiaulys, J.: On a closure property of convolution equivalent class of distributions. J. Math. Anal. Appl. 490, Art. 124226 (2020)

    Google Scholar 

  50. Leslie, J.R.: On the non-closure under convolution of the subexponential family. J. Appl. Probab. 26, 58–66 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  51. Li, J., Tang, Q.: A note on max-sum equivalence. Statist. Probab. Lett. 80, 1720–1723 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  52. Lin, J.X.: Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims. Insurance Math. Econom. 51, 422–429 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. Lin, J., Wang, Y.: New examples of heavy-tailed O-subexponential distributions and related closure properties. Statist. Probab. Lett. 82, 427–432 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  54. Liu, Q., Mao, T., Hu, T.: Closure properties of the second-order regular variation under convolutions. Commun. Stat.- Theory Methods 46, 104–119 (2017)

    Google Scholar 

  55. Mikosch, T.: Regular Variation, Subexponentiality and Their Applications in Probability Theory. Technical Report 99-013, University of Groningen, Groningen (1999)

    Google Scholar 

  56. Mikosch, T.: Non-Life Insurance Mathematics: An Introduction with the Poisson Process. Springer, New York (2009)

    Book  MATH  Google Scholar 

  57. Mikosch, T., Nagaev, A.V.: Large deviations of heavy-tailed sums with applications in insurance. Extremes 1 81–110 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  58. Ng, K.W., Tang, Q.H., Yang, H.: Maxima of sums of heavy-tailed random variables. ASTIN Bulletin 32, 43–55 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  59. Omey, E.: On the difference between the product and the convolution product of distribution functions. Publications de l’Institut Mathématique 55, 111–145 (1994)

    MathSciNet  MATH  Google Scholar 

  60. Pakes, A.G.: Convolution equivalence and infinite divisibility. J. Appl. Probab. 41, 407–424 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  61. Pitman, E.J.G.: Subexponential distribution functions. J. Aust. Math. Soc. Ser. A 29, 337–347 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  62. Rogozin, B.A., Sgibnev, M.S.: Banach algebras of measures on the line with given asymptotics of distributions at infinity. Sib. Math. J. 40, 565–576 (1999) (Sibirskii Matematicheskii Zhurnal 40(3), 660–672 (1999))

    Google Scholar 

  63. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.L.: Stochastic Processes for Insurance and Finance. Wiley, Chichester (1999)

    Book  MATH  Google Scholar 

  64. Samorodnitsky, G.: Stochastic Processes and Long Range Dependence. Springer, New York (2016)

    Book  MATH  Google Scholar 

  65. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York (1994)

    MATH  Google Scholar 

  66. Schmidli H.: Risk Theory. Springer, New York (2017)

    Book  MATH  Google Scholar 

  67. Shimura, T.: Decomposition problem of probability measures related to monotone regularly varying functions. Nagoya Math. J. 135, 87–111 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  68. Sprindys, J., Šiaulys, J.: Regularly distributed randomly stopped sum, minimum, and maximum. Nonlinear Anal. Modell. Control 25, 509–522 (2020)

    MathSciNet  MATH  Google Scholar 

  69. Su, C., Jiang T., Tang Q.: Extension of some classical results on ruin probability to delayed renewal model. Acta Math. Sin. Engl. Ser. 18, 675–680 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  70. Tang, Q., Tsitsiashvili, G.: Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Processes Appl. 108, 299–325 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  71. Tang, Q., Yan, J.: A sharp inequality for the tail probabilities of sums of i.i.d. r.v.’s with dominatedly varying tails. Sci. China Ser. A 45, 1006–1011 (2002)

    MATH  Google Scholar 

  72. Teugels, J.L.: The class of subexponential distributions. Ann. Probab. 3, 1000–1011 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  73. Veraverbeke, N.: Asymptotic behaviour of Wiener-Hopf factors of a random walk. Stochastic Processes Appl. 5, 27–37 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  74. Wang, S., Guo, D., Wang, W.: Closure property of consistently varying random variables based on precise large deviation principles. Commun. Stat.- Theory Methods 48, 2218–2228 (2019)

    Google Scholar 

  75. Wang, Y., Xu, H., Cheng, D., Yu, C.: The local asymptotic estimation for the supremum of a random walk with generalized strong subexponential summands. Stat. Pap. 59, 99–126 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  76. Wang, Y., Yin, C.: Minimum of dependent random variables with convolution-equivalent distributions. Commun. Stat.- Theory Methods 40, 3245–3251 (2011)

    Google Scholar 

  77. Watanabe, T.: Convolution equivalence and distributions of random sums. Probab. Theory Relat. Fields 142, 367–397 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  78. Watanabe, T.: Subexponential densities of infinitely divisible distributions on the half-line. Lith. Math. J. 60, 530–543 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  79. Watanabe, T., Yamamuro, K.: Ratio of the tail of an infinitely divisible distribution on the line to that of its Lévy measure. Electron. J. Probab. 15, 44–74 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  80. Watanabe, T., Yamamuro, K.: Local subexponentiality and self-decomposability. J. Theor. Probab. 23, 1039–1067 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  81. Watanabe, T., Yamamuro, K.: Two non-closure properties on the class of subexponential densities. J. Theor. Probab. 30, 1059–1075 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  82. Willekens, E.: The structure of the class of subexponential distributions. Probab. Theory Relat. Fields 77, 567–581 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  83. Willmot, G.E., Lin, X.S.: Lundberg Approximations for Compound Distributions with Insurance Applications. Lecture Notes in Statistics, vol. 156. Springer, New York (2001)

    Google Scholar 

  84. Willmot, G.E., Woo, J.-K.: Surplus Analysis of Sparre Andersen Insurance Risk Processes. Springer, New York (2017)

    Book  MATH  Google Scholar 

  85. Yakymiv, A.L.: Sufficient conditions for the subexponential property of the convolution of two distributions. Matematicheskie Zametki 58(5), 778–781 (1995) (Math. Notes 58, 1227–1230 (1995))

    Google Scholar 

  86. Yakymiv, A.L.: Some properties of subexponential distributions. Matematicheskie Zametki 62(1), 138–144 (1997) (Math. Notes 62, 116–121 (1997))

    Google Scholar 

  87. Yang, Y., Yuen, K.C., Liu, J.-F.: Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims. J. Ind. Manage. Optim. 14, 231–247 (2018)

    Article  MATH  Google Scholar 

  88. Yu, C., Wang, Y.: Tail behavior of supremum of a random walk when Cramér’s condition fails. Frontiers of Mathematics in China 9, 431–453 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  89. Yu, C., Wang, Y., Cheng, D.: Tail behavior of the sums of dependent and heavy-tailed random variables. J. Korean Stat. Soc. 44, 12–27 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  90. Yu, C., Wang, Y., Cui, Z.: Lower limits and upper limits for tails of random sums supported on \(\mathbf {R}\). Statist. Probab. Lett. 80, 1111–1120 (2010)

    Google Scholar 

  91. Yu, C., Wang, Y., Yang, Y.: The closure of the convolution equivalent distribution class under convolution roots with application to random sums. Statist. Probab. Lett. 80, 462–472 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  92. Yuen, K.C., Yin, C.: Asymptotic results for tail probabilities of sums of dependent and heavy-tailed random variables. Chin. Ann. Math. Ser. B 33, 557–568 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  93. Zachary, S.: A note on Veraverbeke’s theorem. Queueing Syst. 46, 9–14 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  94. Zachary, S., Foss, S.G.: On the exact distributional asymptotics for the supremum of a random walk with increments in a class of light-tailed distributions. Sib. Math. J. 47, 1034–1041 (2006)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics, Vilnius University, Vilnius, Lithuania

    Remigijus Leipus

  2. Institute of Mathematics, Vilnius University, Vilnius, Lithuania

    Jonas Šiaulys

  3. Department of Statistics and Actuarial - Financial Mathematics, University of the Aegean, Karlovassi, Greece

    Dimitrios Konstantinides

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  1. Remigijus Leipus
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  2. Jonas Šiaulys
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  3. Dimitrios Konstantinides
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Leipus, R., Šiaulys, J., Konstantinides, D. (2023). Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing, Maximum, and Minimum. In: Closure Properties for Heavy-Tailed and Related Distributions. SpringerBriefs in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-031-34553-1_3

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