Abstract
Let X 1, … , X n be independent nonnegative random variables with respective survival functions \(\overline {F}_{1}, \ldots , \overline {F}_{n}\), and let Θ1, … , Θ n be (not necessarily independent) nonnegative random variables, independent of X 1, … , X n , satisfying certain moment conditions. This paper consists of two parts. In the first part, we investigate second-order expansions of 𝗣 \( \left ({\sum }^{n}_{i=1} X_{i}>t\right )\) as \(t\to {\infty }\) under the assumption that the \(\overline {F}_{i}\) are of second-order regular variation (2RV) with the same first-order index but with different second-order indexes. In the second part, under the assumption that the \(\overline {F}_{1}=\cdots =\overline {F}_{n}\) have 2RV tails, second-order expansions of tail probabilities of the randomly weighted sum \({\sum }^{n}_{i=1} {\Theta }_{i} X_{i}\) are studied. The closure property of 2RV under randomly weighted sum is also discussed. The main results in this paper generalize and strengthen several known results in the literature.
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References
Barbe, P., McCormick, W.P.: Asymptotic expansions of convolutions of regularly varying distributions. J. Aust. Math. Soc. 78, 339–371 (2005)
Barbe, P., McCormick, W.P.: Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications. Memoirs of the American Mathematical Society 197(922), 124 (2009)
Basrak, B., Davis, R.A., Mikosch, T.: Regular variation of GARCH processes. Stoch. Process. Appl. 99(1), 95–115 (2002)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)
de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)
de Haan, L., Resnick, S.: Second-order regular variation and rates of convergence in extreme-value theory. Ann. Probab., 97–124 (1996)
Degen, M., Lambrigger, D.D., Segers, J.: Risk concentration and diversification: Second-order properties. Insurance: Mathematics and Economics 46, 541–546 (2010)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Finance and Insurance. Springer, Berlin (1997)
Feller, W.: An Introduction to Probability Theory and its Applications, vol. 2nd edn. II. Wiley, New York (1971)
Geluk, J.L.: Tails of subordinated laws: the regularly varying case. Stoch. Process. Appl. 61, 147–161 (1996)
Geluk, J.L, de Haan, L., Resnick, S., Stǎricǎ, C.: Second-order regular variation, convolution and the central limit theorem. Stoch. Process. Appl. 69, 139–159 (1997)
Geluk, J.L., Peng, L.: An adaptive optimal estimate of the tail index for MA(1) time series. Statistics and Probability Letters 46, 217–227 (2000)
Geluk, J.L., Peng, L., De Vries, C.: Convolutions of heavy-tailed random variables and applications to portfolio diversification and MA(1) time series. Adv. Appl. Probab. 32, 1011–1026 (2000)
Heffernan, J., Resnick, S.: Hidden regular variation and the rank transform. Adv. Appl. Probab. 37, 393–414 (2005)
Hua, L., Joe, H.: Second order regular variation and conditional tail expectation of multiple risks. Insurance Mathematics and Economics 49, 537–546 (2011)
Liu, Q., Mao, T., Hu, T.: Closure properties of the second-order regular variation under Convolutions. Preprint. University of Science and Technology of China (2015)
Mao, T., Hu, T.: Second-order properties of risk concentrations without the condition of asymptotic smoothness. Extremes 16(4), 383–405 (2013)
Mao, T.: Second-order conditions of regular variation and inequalities of Drees type. In: Li, H., Li, X. (eds.) Lecture Notes in Statistics in Honor of Professor Moshe Shaked on his 70th Birthday, pp 233–246. Springer (2013)
Mao, T., Lv, W., Hu, T.: Second-order expansions of risk concentration based on CTE. Insurance: Mathematics and Economics 51, 449–456 (2012)
Maulik, K., Resnick, S.: Characterizations and examples of hidden regular variation. Extremes 7, 31–67 (2004)
Mitra, A., Resnick, S.: Hidden regular variation: detection and estimation (2010). arXiv:1001.5058
Resnick, S.I.: Point processes, regular variation and weak convergence. Adv. Appl. Probab. 18, 66–138 (1986)
Resnick, S.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987)
Resnick, S.: Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5, 303–336 (2002)
Resnick, S.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007)
Resnick, S.I., Willekens, E.: Moving averages with random coefficients and random coefficient autoregressive models. Stoch. Model. 7(4), 511–525 (1991)
Zhang, Y., Shen, X., Weng, C.: Approximation of tail probability of randomly weighted sums and applications. Stoch. Process. Appl. 119(2), 655–675 (2009)
Zhu, L., Li, H.: Asymptotic analysis of multivariate tail conditional expectations. North American Actuarial Journal 16(3), 350–363 (2012)
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Mao, T., Ng, K.W. Second-order properties of tail probabilities of sums and randomly weighted sums. Extremes 18, 403–435 (2015). https://doi.org/10.1007/s10687-015-0218-0
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DOI: https://doi.org/10.1007/s10687-015-0218-0