Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T10:40:02.808Z Has data issue: false hasContentIssue false
  • English
  • Français

A non-exponential extension of Sanov’s theorem via convex duality

Part of: Limit theorems Miscellaneous applications of functional analysis

Published online by Cambridge University Press:  29 April 2020

Daniel Lacker
Daniel Lacker*
Affiliation:
Columbia University
*
*Postal address: 306 Mudd, 500 West 120th St, New York, NY 10027, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This work is devoted to a vast extension of Sanov’s theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis–Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

Keywords

Sanov’s theoremlarge deviationsconvex dualityrisk measuresweak convergenceempirical measuresheavy tailsstochastic optimization

MSC classification

Primary: 60F10: Large deviations
Secondary: 46N10: Applications in optimization, convex analysis, mathematical programming, economics
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 1 , March 2020 , pp. 61 - 101
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acciaio, B. andPenner, I. (2011). Dynamic risk measures. In Advanced Mathematical Methods for Finance, eds Di Nunno, G. andØksendal, B., pp. 1–34. Springer.Google Scholar
Agueh, M. andCarlier, G. (2011). Barycenters in the Wasserstein space. SIAM J. Math. Anal. 43 (2), 904–924.CrossRefGoogle Scholar
Aliprantis, C. andBorder, K. (2007). Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer.Google Scholar
Atar, R., Chowdhary, K. andDupuis, P. (2015). Robust bounds on risk-sensitive functionals via Rényi divergence. SIAM/ASA J. Uncertain. Quantif. 3 (1), 18–33.CrossRefGoogle Scholar
Backhoff-Veraguas, J., Lacker, D. andTangpi, L. (2018). Non-exponential Sanov and Schilder theorems on Wiener space: BSDEs, Schrödinger problems and control. Available at arXiv:1810.01980 .Google Scholar
Bartl, D. (2019). Conditional nonlinear expectations. Stoch. Process Appl. arXiv:1612.09103v2Google Scholar
Ben-Tal, A. andTeboulle, M. (1986). Expected utility, penalty functions, and duality in stochastic nonlinear programming. Manag. Sci. 32 (11), 1445–1466.CrossRefGoogle Scholar
Ben-Tal, A. andTeboulle, M. (2007). An old–new concept of convex risk measures: the optimized certainty equivalent. Math. Finance 17 (3), 449–476.CrossRefGoogle Scholar
Bertsekas, D. andShreve, S. (1996). Stochastic Optimal Control: The Discrete Time Case. Athena Scientific.Google Scholar
Blanchet, A. andCarlier, G. (2015). Optimal transport and Cournot–Nash equilibria. Math. Operat. Res. 41 (1), 125–145.CrossRefGoogle Scholar
Bobkov, S. andDing, Y. (2014). Optimal transport and Rényi informational divergence. Preprint.Google Scholar
Boissard, E. (2011). Simple bounds for the convergence of empirical and occupation measures in 1-Wasserstein distance. Electron. J. Prob. 16, 2296–2333.CrossRefGoogle Scholar
Borovkov, A. A. andBorovkov, K. A. (2008). Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions (Encyclopedia Math. Appl. 118). Cambridge University Press.Google Scholar
Cheridito, P. andKupper, M. (2011). Composition of time-consistent dynamic monetary risk measures in discrete time. Internat. J. Theoret. Appl. Finance 14 (1), 137–162.CrossRefGoogle Scholar
De Haan, L. andLin, T. (2001). On convergence toward an extreme value distribution in C[0, 1]. Ann. Prob. 29, 467–483.CrossRefGoogle Scholar
Dembo, A. andZeitouni, O. (2009). Large Deviations Techniques and Applications (Stoch. Model. Appl. Prob. 38). Springer Science & Business Media.Google Scholar
Denisov, D., Dieker, A. B. andShneer, V. (2008). Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Prob. 36 (5), 1946–1991.CrossRefGoogle Scholar
Ding, Y. (2014). Wasserstein-Divergence transportation inequalities and polynomial concentration inequalities. Statist. Prob. Lett. 94, 77–85.CrossRefGoogle Scholar
Dudley, R. M. (1969). The speed of mean Glivenko–Cantelli convergence. Ann. Math. Statist. 40 (1), 40–50.CrossRefGoogle Scholar
Dudley, R. M. (2018). Real Analysis and Probability. CRC Press.Google Scholar
Dupacová, J. andWets, R. J. B. (1988). Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems. Ann. Statist. 16, 1517–1549.CrossRefGoogle Scholar
Dupuis, P. andEllis, R. S. (2011). A Weak Convergence Approach to the Theory of Large Deviations (Wiley Series Prob. Statist. 902). John Wiley.Google Scholar
Eckstein, S. (2019). Extended Laplace principle for empirical measures of a Markov chain. Adv. Appl. Prob. 51 (1), 136–167. arXiv:1709.02278CrossRefGoogle Scholar
Eichelsbacher, P. andSchmock, U. (1996). Large deviations of products of empirical measures and U-empirical measures in strong topologies. Sonderforschungsbereich 343, Diskrete Strukturen in der Math., Universität Bielefeld.Google Scholar
Einmahl, U. andLi, D. (2008). Characterization of LIL behavior in Banach space. Trans. Amer. Math. Soc. 360 (12), 6677–6693.CrossRefGoogle Scholar
Föllmer, H. andKnispel, T. (2011). Entropic risk measures: coherence vs. convexity, model ambiguity and robust large deviations. Stoch. Dynamics 11 (02n03), 333–351.Google Scholar
Föllmer, H. andSchied, A. (2002). Convex measures of risk and trading constraints. Finance Stochast. 6 (4), 429–447.CrossRefGoogle Scholar
Föllmer, H. andSchied, A. (2011). Stochastic Finance: An Introduction in Discrete Time. Walter de Gruyter.CrossRefGoogle Scholar
Foss, S., Korshunov, D. andZachary, S. (2011). An Introduction to Heavy-Tailed and Subexponential Distributions (Springer Series Operat. Research Financial Eng. 6). Springer.Google Scholar
Fournier, N. andGuillin, A. (2015). On the rate of convergence in Wasserstein distance of the empirical measure. Prob. Theory Relat. Fields 162 (3–4), 707–738.CrossRefGoogle Scholar
Fuqing, G. andMingzhou, X. (2012). Relative entropy and large deviations under sublinear expectations. Acta Math. Sci. 32 (5), 1826–1834.CrossRefGoogle Scholar
Hardy, G. H., Littlewood, J. E. andPólya, G. (1952). Inequalities. Cambridge University Press.Google Scholar
Hu, F. (2010). On Cramér’s theorem for capacities. Comptes Rendus Mathématique 348 (17), 1009–1013.CrossRefGoogle Scholar
Hult, H. andLindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (NS) 80 (94), 121–140.CrossRefGoogle Scholar
Hult, H., Lindskog, F., Mikosch, T. andSamorodnitsky, G. (2005). Functional large deviations for multivariate regularly varying random walks. Ann. Appl. Prob. 15 (4), 2651–2680.CrossRefGoogle Scholar
Kall, P. (1987). On Approximations and Stability in Stochastic Programming. In Parametric Optimization and Related Topics, eds Guddat, J.et al. pp. 387–407. Akademie, Berlin.Google Scholar
Kaniovski, Y. M., King, A. J. andWets, R. J. B. (1995). Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems. Ann. Operat. Res. 56 (1), 189–208.CrossRefGoogle Scholar
King, A. J. andWets, R. J. B. (1991). Epi-consistency of convex stochastic programs. Stoch. Stoch. Reports 34 (1–2), 83–92.CrossRefGoogle Scholar
Lacker, D. (2018). Liquidity, risk measures, and concentration of measure. Math. Operat. Res. 34 (3), 693–1050, C2.arXiv:1510.07033CrossRefGoogle Scholar
Lindskog, F., Resnick, S. I. andRoy, J. (2014). Regularly varying measures on metric spaces: hidden regular variation and hidden jumps. Prob. Surveys 11, 270–314.CrossRefGoogle Scholar
Mikosch, T. andNagaev, A. V. (1998). Large deviations of heavy-tailed sums with applications in insurance. Extremes 1 (1), 81–110.CrossRefGoogle Scholar
Mogul’skii, A. A. (1977). Large deviations for trajectories of multi-dimensional random walks. Theory Prob. Appl. 21 (2), 300–315.CrossRefGoogle Scholar
Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7 (5), 745–789.CrossRefGoogle Scholar
Owari, K. (2014). Maximum Lebesgue extension of monotone convex functions. J. Funct. Anal. 266 (6), 3572–3611.CrossRefGoogle Scholar
Parthasarathy, K. R. (2005). Probability Measures on Metric Spaces (Prob. Math. Statist. 352). American Mathematical Society.Google Scholar
Peng, S. (2007). Law of large numbers and central limit theorem under nonlinear expectations. Available at arXiv:math/0702358 .Google Scholar
Peng, S. (2010). Nonlinear expectations and stochastic calculus under uncertainty. Available at arXiv:1002.4546 .Google Scholar
Petrov, V. (2012). Sums of Independent Random Variables. Springer Science & Business Media.Google Scholar
Rhee, C.-H., Blanchet, J. andZwart, B. (2016). Sample path large deviations for heavy-tailed Lévy processes and random walks. Available at arXiv:1606.02795 .Google Scholar
Schied, A. (1998). Cramer’s condition and Sanov’s theorem. Statist. Prob. Lett. 39 (1), 55–60.CrossRefGoogle Scholar
Sion, M. (1958). On general minimax theorems. Pacific J. Math. 8 (1), 171–176.CrossRefGoogle Scholar
Villani, C. (2003). Topics in Optimal Transportation (Graduate Studies Math. 58). American Mathematical Society.Google Scholar
Wang, R., Wang, X. andWu, L. (2010). Sanov’s theorem in the Wasserstein distance: a necessary and sufficient condition. Statist. Prob. Lett. 80 (5), 505–512.CrossRefGoogle Scholar
Weed, J. andBach, F. (2019). Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance. Bernoulli 25 (4A), 2620–2648.arXiv:1707.00087CrossRefGoogle Scholar
Zalinescu, C. (2002). Convex Analysis in General Vector Spaces. World Scientific.CrossRefGoogle Scholar