American Institute of Mathematical Sciences

March  2021, 6(1): 53-60. doi: 10.3934/puqr.2021003

Improved Hoeffding inequality for dependent bounded or sub-Gaussian random variables

 Institute for Business and Finance, Waseda University, 3rd Floor, Bldg.11 1-6-1 Nishi-Waseda, Shinjuku-ku, 169-8050 Tokyo, JP Full list of author information is available at the end of the article

Email: tanoue.yuta@aoni.waseda.jp

Received  January 30, 2020 Accepted  January 06, 2021 Published  March 2021

Fund Project: This work was supported by JSPS Grant-in-Aid for Young Scientists(Grant No.18K12873) and Waseda University Grants for Special Research Projects(“Tokutei Kadai”) (Grant No. 2019C-688). The authour thanks the anonymous referee and the associate editor for their careful reading of the paper and constructive comments.

When addressing various financial problems, such as estimating stock portfolio risk, it is necessary to derive the distribution of the sum of the dependent random variables. Although deriving this distribution requires identifying the joint distribution of these random variables, exact estimation of the joint distribution of dependent random variables is difficult. Therefore, in recent years, studies have been conducted on the bound of the sum of dependent random variables with dependence uncertainty. In this study, we obtain an improved Hoeffding inequality for dependent bounded variables. Further, we expand the above result to the case of sub-Gaussian random variables.

Citation: Yuta Tanoue. Improved Hoeffding inequality for dependent bounded or sub-Gaussian random variables. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 53-60. doi: 10.3934/puqr.2021003
References:
 [1] George Bennett, Probability inequalities for the sum of independent random variables, Journal of the American Statistical Association, 1962, 57(297): 33-45. doi: 10.1080/01621459.1962.10482149. [2] Carole Bernard, Ludger Rüschendorf, Steven Vanduffel and Ruodu Wang, Risk bounds for factor models, Finance and Stochastics, 2017, 21(3): 631-659. doi: 10.1007/s00780-017-0328-4. [3] Carole Bernard, Ludger Rüschendorf, Steven Vanduffel and Jing Yao, How robust is the value-at-risk of credit risk portfolios? The European Journal of Finance, 2017, 23(6): 507-534. doi: 10.1080/1351847X.2015.1104370. [4] Sergei Bernstein, The Theory of Probabilities (in Russian), Gastehizdat Publishing House, 1946. [5] Philip Best, Implementing Value at Risk, Wiley, 2010. [6] Valeria Bignozzi, Giovanni Puccetti, and Ludger Rüschendorf, Reducing model risk via positive and negative dependence assumptions, Insurance: Mathematics and Economics, 2015, 61: 17-26. doi: 10.1016/j.insmatheco.2014.11.004. [7] Richard C Bradley, Approximation theorems for strongly mixing random variables, The Michigan Mathematical Journal, 1983, 30(1): 69-81. doi: 10.1307/mmj/1029002789. [8] Umberto Cherubini, Elisa Luciano and Walter Vecchiato, Copula Methods in Finance, John Wiley & Sons, 2004. [9] Moorad Choudhry and Max Wong, An Introduction to Value-at-Risk, John Wiley & Sons, 2013. [10] Paul Embrechts, Filip Lindskog, and Alexander McNeil, Modelling Dependence with Copulas. Rapport technique, Département de Mathématiques, Institut Fédéral de Technologie de Zurich, Zurich, 2001. [11] Paul Embrechts and Giovanni Puccetti, Bounds for functions of dependent risks, Finance and Stochastics, 2006, 10(3): 341-352. doi: 10.1007/s00780-006-0005-5. [12] Edward W Frees and Emiliano A Valdez, Understanding relationships using copulas, North American Actuarial Journal, 1998, 2(1): 1-25. doi: 10.1080/10920277.1998.10595667. [13] Christian Genest, Michel Gendron and Michaël Bourdeau-Brien, The advent of copulas in finance. In: Copulae and Multivariate Probability Distributions in Finance, Routledge, 2013: 13-25. [14] Wassily Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association, 1963, 58(301): 13-30. doi: 10.1080/01621459.1963.10500830. [15] Akira Ieda, Kohei Marumo, Toshinao Yoshiba, A Simplified Method for Calculating the Credit Risk of Lending Portfolios. Institute for Monetary and Economic Studies, Bank of Japan, 2000. [16] Svante Janson, Large deviations for sums of partly dependent random variables, Random Structures & Algorithms, 2004, 24(3): 234-248. [17] Christoph H Lampert, Liva Ralaivola and Alexander Zimin, Dependency-dependent bounds for sums of dependent random variables. arXiv: 1811.01404, 2018. [18] Colin McDiarmid, On the method of bounded differences, Surveys in Combinatorics, 1989, 141(1): 148-188. [19] Magda Peligrad, Some remarks on coupling of dependent random variables, Statistics & Probability Letters, 2002, 60(2): 201-209. [20] Giovanni Puccetti, Ludger Rüschendorf, Daniel Small and Steven Vanduffel, Reduction of value-at-risk bounds via independence and variance information, Scandinavian Actuarial Journal, 2017, 2017(3): 245-266. [21] Emmanuel Rio, Inequalities and limit theorems for weakly dependent sequences, Lecture, 2013. [22] Ludger Rüschendorf, Risk bounds and partial dependence information, In: From Statistics to Mathematical Finance, 2017: 345-366.

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References:
 [1] George Bennett, Probability inequalities for the sum of independent random variables, Journal of the American Statistical Association, 1962, 57(297): 33-45. doi: 10.1080/01621459.1962.10482149. [2] Carole Bernard, Ludger Rüschendorf, Steven Vanduffel and Ruodu Wang, Risk bounds for factor models, Finance and Stochastics, 2017, 21(3): 631-659. doi: 10.1007/s00780-017-0328-4. [3] Carole Bernard, Ludger Rüschendorf, Steven Vanduffel and Jing Yao, How robust is the value-at-risk of credit risk portfolios? The European Journal of Finance, 2017, 23(6): 507-534. doi: 10.1080/1351847X.2015.1104370. [4] Sergei Bernstein, The Theory of Probabilities (in Russian), Gastehizdat Publishing House, 1946. [5] Philip Best, Implementing Value at Risk, Wiley, 2010. [6] Valeria Bignozzi, Giovanni Puccetti, and Ludger Rüschendorf, Reducing model risk via positive and negative dependence assumptions, Insurance: Mathematics and Economics, 2015, 61: 17-26. doi: 10.1016/j.insmatheco.2014.11.004. [7] Richard C Bradley, Approximation theorems for strongly mixing random variables, The Michigan Mathematical Journal, 1983, 30(1): 69-81. doi: 10.1307/mmj/1029002789. [8] Umberto Cherubini, Elisa Luciano and Walter Vecchiato, Copula Methods in Finance, John Wiley & Sons, 2004. [9] Moorad Choudhry and Max Wong, An Introduction to Value-at-Risk, John Wiley & Sons, 2013. [10] Paul Embrechts, Filip Lindskog, and Alexander McNeil, Modelling Dependence with Copulas. Rapport technique, Département de Mathématiques, Institut Fédéral de Technologie de Zurich, Zurich, 2001. [11] Paul Embrechts and Giovanni Puccetti, Bounds for functions of dependent risks, Finance and Stochastics, 2006, 10(3): 341-352. doi: 10.1007/s00780-006-0005-5. [12] Edward W Frees and Emiliano A Valdez, Understanding relationships using copulas, North American Actuarial Journal, 1998, 2(1): 1-25. doi: 10.1080/10920277.1998.10595667. [13] Christian Genest, Michel Gendron and Michaël Bourdeau-Brien, The advent of copulas in finance. In: Copulae and Multivariate Probability Distributions in Finance, Routledge, 2013: 13-25. [14] Wassily Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association, 1963, 58(301): 13-30. doi: 10.1080/01621459.1963.10500830. [15] Akira Ieda, Kohei Marumo, Toshinao Yoshiba, A Simplified Method for Calculating the Credit Risk of Lending Portfolios. Institute for Monetary and Economic Studies, Bank of Japan, 2000. [16] Svante Janson, Large deviations for sums of partly dependent random variables, Random Structures & Algorithms, 2004, 24(3): 234-248. [17] Christoph H Lampert, Liva Ralaivola and Alexander Zimin, Dependency-dependent bounds for sums of dependent random variables. arXiv: 1811.01404, 2018. [18] Colin McDiarmid, On the method of bounded differences, Surveys in Combinatorics, 1989, 141(1): 148-188. [19] Magda Peligrad, Some remarks on coupling of dependent random variables, Statistics & Probability Letters, 2002, 60(2): 201-209. [20] Giovanni Puccetti, Ludger Rüschendorf, Daniel Small and Steven Vanduffel, Reduction of value-at-risk bounds via independence and variance information, Scandinavian Actuarial Journal, 2017, 2017(3): 245-266. [21] Emmanuel Rio, Inequalities and limit theorems for weakly dependent sequences, Lecture, 2013. [22] Ludger Rüschendorf, Risk bounds and partial dependence information, In: From Statistics to Mathematical Finance, 2017: 345-366.
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