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A Journal of Theoretical and Applied Statistics
Volume 56, 2022 - Issue 1
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Research Article

Central limit theorem and almost sure results for the empirical estimator of superquantiles/CVaR in the stationary case

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Pages 53-72 | Received 08 May 2021, Accepted 14 Feb 2022, Published online: 28 Feb 2022

References

  • Acerbi C, Tasche D. On the coherence of expected shortfall. J Bank Finance. 2002;26:1487–1503.
  • Rockafellar RT, Uryasev S. Optimization of conditional value-at-risk. J Risk. 2000;2:21–41.
  • Rachev ST, Stoyanov S, Fabozzi FJ. Advanced stochastic models, risk assessment, and portfolio optimization: the ideal risk, uncertainty, and performance measures. Hoboken: John Wiley; 2007.
  • Rockafellar RT, Royset JO. Superquantiles, and their applications to risk, random variables, and regression. In: Tutorials in operation research. INFORMS; 2013. p. 151–167.
  • Artzner P, Delbaen F, Eber J-M, et al. Coherent measure of risk. Math Finance. 1999;9(3):203–228.
  • Pflug G. Some remarks on the value-at-risk and the conditional value-at-risk. In: Uryasev S, editor. Probabilistic constrained optimization: methodology and applications. Dordrecht: Kluwer; 2000.
  • Trindade AA, Uryasev S, Shapiro A, et al. Financial prediction with constrained tail risk. J Bank Finance. 2007;31:3524–3538.
  • Nagaraja H. Some nondegenerate limit laws for the selection differential. Ann Stat. 1982;10(4):1306–1310.
  • Labopin-Richard T, Gamboa F, Garivier A, et al. Bregman superquantiles. Estimation methods and applications. Depend Model. 2016;4(1):76–108.
  • Rosenblatt M. A central limit theorem and a strong mixing condition. Proc Natl Acad Set USA. 1956;42:43–47.
  • Dedecker J, Gouëzel S, Merlevède F. Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains. Ann Inst Henri Poincaré Probab Stat. 2010;46(3):796–821.
  • Doukhan P, Massart P, Rio E. The functional central limit theorem for strongly mixing processes. Ann Inst H Poincaré Probab Stat. 1994;30(1):63–82.
  • Bradley RC. On quantiles and the central limit question for strongly mixing sequences. Dedicated to Murray Rosenblatt. J Theor Probab. 1997;10(2):507–555.
  • Rio E. Asymptotic theory of weakly dependent random processes. Translated from the 2000 French edition. Probability theory and stochastic modelling. Vol. 80. Berlin: Springer; 2017. xviii+204 pp.
  • Dedecker J, Prieur C. An empirical central limit theorem for dependent sequences. Stoch Process Appl. 2007;117(1):121–142.
  • Dedecker J, Merlevède F. Almost sure invariance principle for the Kantorovich distance between the empirical and the marginal distributions of strong mixing sequences. Stat Probab Lett. 2021;171:108991.
  • Dedecker J, Rio E. On the functional central limit theorem for stationary processes. Ann Inst H Poincaré Probab Stat. 2000;36(1):1–34.
  • Liverani C, Saussol B, Vaienti S. A probabilistic approach to intermittency. Ergodic Theory Dyn Syst. 1999;19(3):671–685.
  • Gouëzel S. Central limit theorem and stable laws for intermittent maps. Probab Theory Related Fields. 2004;128(1):82–122.
  • Berthet P, Dedecker J, Merlevède F. Central limit theorem and almost sure results for bivariate empirical W1 distances. Pub Inst Stat Univ Paris. 2019;63(fasc. 2-3):205–220.
  • Rio E. A maximal inequality and dependent Marcinkiewicz-Zygmund strong laws. Ann Probab. 1995;23(2):918–937.
  • Dedecker J, Dehling H, Taqqu MS. Weak convergence of the empirical process of intermittent maps in L2 under long-range dependence. Stoch Dyn. 2015;15(2):29.
  • Cuny C. Invariance principles under the Maxwell-Woodroofe condition in Banach spaces. Ann Probab. 2017;45:1578–1611.
  • Dedecker J. A central limit theorem for stationary random fields. Probab Theory Related Fields. 1998;110(3):397–426.
  • Andrews DWK. Non-strong mixing autoregressive processes. J Appl Probab. 1984;21(4):930–934.
  • Caron E, Dedecker J, Michel B. Linear regression with stationary errors: the R package slm. R J. 2021;13(1):83–100.

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