Precise large deviations for sums of random vectors in a multidimensional size-dependent renewal risk model

  • Xin-mei ShenEmail author
  • Ke-ang Fu
  • Xue-ting Zhong


Consider a multidimensional renewal risk model, in which the claim sizes {Xk, k ≥ 1} form a sequence of independent and identically distributed random vectors with nonnegative components that are allowed to be dependent on each other. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Suppose that the claim sizes and inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure. A precise large deviation for the multidimensional renewal risk model is obtained.


Precise large deviation Size-dependent Consistent variation Multidimensional risk model Renewal counting process 

MR Subject Classification

60F10 60G50 60K05 62P05 62H99 91B30 


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  1. [1]
    E Sparre Andersen. On the collective theory of risk in case of contagion between claims, Transactions of the XVth International Congress of Actuaries, New York, 1957, 2: 219–229.Google Scholar
  2. [2]
    AV Asimit, AL Badescu. Extremes on the discounted aggregate claims in a time dependent risk model, Scand Actuar J, 2010, 2010(2): 3–104.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    AL Badescu, ECK Cheung, D Landriault. Dependent risk models with bivariate phase-type distributions, J Appl Probab, 2009, 46(1): 113–131.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A Baltrūnas, R Leipus, J Šiaulys. Dependent risk models with bivariate phase-type distributions, Precise large deviation results for the total claim amount under subexponential claim sizes, 2008, 78(10): 1206–1214.Google Scholar
  5. [5]
    X Bi, S Zhang. Precise large deviations of aggregate claims in a risk model with regression-type size-dependence, Stat Probab Lett, 2013, 83(10): 2248–2255.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    NH Bingham, CM Goldie, J LTeugels. Regular variation, Cambridge University Press, 1987, Cambridge.CrossRefGoogle Scholar
  7. [7]
    Y Chen, KC Yuen. Precise large deviations of aggregate claims in a size-dependent renewal risk model, Insurance Math Econom, 2012, 51(2): 457–461.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Y Chen, KC Yuen, KW Ng. Asymptotics for the ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims, Appl Stoch Models Bus Ind, 2011, 27: 290–300.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    DB HCline, G Samorodnitsky. Subexponentiality of the product of independent random variables, Stochastic Process Appl, 1994, 49(1): 75–98.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    H Cossette, E Marceau, F Marri. On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula, Insurance Math Econom, 2008, 43(3): 444–455.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    P Embrechts, C Klüppelberg, T Mikosch. Modelling Extremal Events: for Insurance and Finance, Springer, 1997, Berlin.CrossRefzbMATHGoogle Scholar
  12. [12]
    J Feng, P Zhao, L Jiao. Local precise large deviations for independent sums in multi-risk model, J Math Res Appl, 2014, 34(2): 240–248.MathSciNetzbMATHGoogle Scholar
  13. [13]
    K Fu, X Shen. Moderate deviations for sums of dependent claims in a size-dependent renewal risk model, Comm Statist Theory Methods, 2017, 46(7): 3235–3243.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    R Kaas, Q Tang. A large deviation result for aggregate claims with dependent claim occurrences, Insurance Math Econom, 2005, 36(3): 251–259.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    C Klüppelberg, T Mikosch. Large deviations of heavy-tailed random sums with applications in insurance and finance, J Appl Probab, 1997, 34(2): 293–308.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    B Ko, Q Tang. Sums of dependent nonnegative random variables with subexponential tails, J Appl Probab, 2008, 45: 85–94.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J Kočetova, R Leipus, J Šiaulys. A property of the renewal counting process with application to the finite-time ruin probability, Lithuanian Math J, 2009, 49(1): 55–61.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    J Li, Q Tang, R Wu. Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Adv Appl Probab, 2010, 42(4): 1126–1146.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    D Lu. Lower bounds of large deviation for sums of long-tailed claims in a multi-risk model, Statist Probab Lett, 2012, 82(7): 1242–1250.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    RB Nelsen. An Introduction to Copulas, Springer, 2006, New York.zbMATHGoogle Scholar
  21. [21]
    KW Ng, Q Tang, J Yan, H Yang. Precise Large Deviations for Sums of Random Variables with Consistently Varying Tails, J Appl Probab, 2004, 41(1): 93–107.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    X Shen, Y Niu, H Tian. Precise large deviations for sums of random vectors with dependent components of consistently varying tails, Front Math China, 2017, 12(3): 711–732.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    X Shen, H Tian. Precise large deviations for sums of two-dimensional random vectors with de-pendent components of heavy tails, Comm Statist Theory Methods, 2016, 45(21): 6357–6368.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    X Shen, M Xu, EF EAtta Mills. Precise large deviation results for sums of subexponential claims in a size-dependent renewal risk model, Statist Probab Lett, 2016, 114: 6–13.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Q Tang, C Su, T Jiang. Large deviations for heavy-tailed random sums in compound renewal model, Statist Probab Lett, 2001, 21(1): 91–100.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    K Wang, Y Wang, Q Gao. Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate, Methodol Comput Appl Probab, 2013, 15: 109–124.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    S Wang, W Wang. Precise large deviations for sums of random variables with consistently varying tails in multi-risk models, J Appl Probab, 2007, 44(4): 889–900.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    S Wang, W Wang. Precise large deviations for sums of random variables with consistent variation in dependent multi-risk models, Comm Statist Theory Methods, 2013, 42(24): 4444–4459.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesDalian University of TechnologyDalianChina
  2. 2.School of Statistics and MathematicsZhejiang Gongshang UniversityHangzhouChina

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