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Second-order regular variation, convolution and the central limit theorem

Author

Listed:
  • Geluk, J.
  • de Haan, L.
  • Resnick, S.
  • Starica, C.

Abstract

Second-order regular variation is a refinement of the concept of regular variation which is useful for studying rates of convergence in extreme value theory and asymptotic normality of tail estimators. For a distribution tail 1 - F which possesses second-order regular variation, we discuss how this property is inherited by 1 - F2 and 1 - F*2. We also discuss the relationship of central limit behavior of tail empirical processes, asymptotic normality of Hill's estimator and second-order regular variation.

Suggested Citation

  • Geluk, J. & de Haan, L. & Resnick, S. & Starica, C., 1997. "Second-order regular variation, convolution and the central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 69(2), pages 139-159, September.
    • Handle: RePEc:eee:spapps:v:69:y:1997:i:2:p:139-159
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    References listed on IDEAS

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    1. J.H.J. Einmahl, 1990. "The empirical distribution function as a tail estimator," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 44(2), pages 79-82, June.
    2. Geluk, J. L., 1996. "Tails of subordinated laws: The regularly varying case," Stochastic Processes and their Applications, Elsevier, vol. 61(1), pages 147-161, January.
    3. Einmahl, J.H.J., 1992. "Limit theorems for tail processes with application to intermediate quantile estimation," Other publications TiSEM 063e51b0-445d-4764-96a2-4, Tilburg University, School of Economics and Management.
    4. Omey, E. & Willekens, E., 1986. "Second order behaviour of the tail of a subordinated probability distribution," Stochastic Processes and their Applications, Elsevier, vol. 21(2), pages 339-353, February.
    5. de Haan, L. & Peng, L., 1997. "Rates of Convergence for Bivariate Extremes," Journal of Multivariate Analysis, Elsevier, vol. 61(2), pages 195-230, May.
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    Cited by:

    1. Jaap Geluk & Liang Peng & Casper G. de Vries, 1999. "Convolutions of Heavy Tailed Random Variables and Applications to Portfolio Diversification and MA(1) Time Series," Tinbergen Institute Discussion Papers 99-088/2, Tinbergen Institute.
    2. Geluk, J. L. & Peng, Liang, 2000. "An adaptive optimal estimate of the tail index for MA(l) time series," Statistics & Probability Letters, Elsevier, vol. 46(3), pages 217-227, February.
    3. Degen, Matthias & Lambrigger, Dominik D. & Segers, Johan, 2010. "Risk concentration and diversification: Second-order properties," Insurance: Mathematics and Economics, Elsevier, vol. 46(3), pages 541-546, June.
    4. Pan, Xiaoqing & Leng, Xuan & Hu, Taizhong, 2013. "The second-order version of Karamata’s theorem with applications," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1397-1403.
    5. Hua, Lei & Joe, Harry, 2011. "Second order regular variation and conditional tail expectation of multiple risks," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 537-546.
    6. Peng, Zuoxiang & Liao, Xin, 2015. "Second-order asymptotics for convolution of distributions with light tails," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 199-208.
    7. A. Dematteo & S. Clémençon, 2016. "On tail index estimation based on multivariate data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(1), pages 152-176, March.
    8. Mao, Tiantian & Yang, Fan, 2015. "Risk concentration based on Expectiles for extreme risks under FGM copula," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 429-439.
    9. Necir, Abdelhakim & Meraghni, Djamel, 2009. "Empirical estimation of the proportional hazard premium for heavy-tailed claim amounts," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 49-58, August.

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