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Riekert A. Convergence rates for empirical measures of Markov chains in dual and Wasserstein distances. Stat Probab Lett 2022. [DOI: 10.1016/j.spl.2022.109605] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
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Convergence Analysis on Data-Driven Fortet-Mourier Metrics with Applications in Stochastic Optimization. SUSTAINABILITY 2022. [DOI: 10.3390/su14084501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
Abstract
Fortet-Mourier (FM) probability metrics are important probability metrics, which have been widely adopted in the quantitative stability analysis of stochastic programming problems. In this study, we contribute to different types of convergence assertions between a probability distribution and its empirical distribution when the deviation is measured by FM metrics and consider their applications in stochastic optimization. We first establish the quantitative relation between FM metrics and Wasserstein metrics. After that, we derive the non-asymptotic moment estimate, asymptotic convergence, and non-asymptotic concentration estimate for FM metrics, which supplement the existing results. Finally, we apply the derived results to four kinds of stochastic optimization problems, which either extend the present results to more general cases or provide alternative avenues. All these discussions demonstrate the motivation as well as the significance of our study.
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Galeati L, Harang FA, Mayorcas A. Distribution dependent SDEs driven by additive continuous noise. ELECTRON J PROBAB 2022. [DOI: 10.1214/22-ejp756] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Lucio Galeati
- Institute of Applied Mathematics, University of Bonn, 53115 Endenicher Allee 60, Bonn, Germany
| | - Fabian A. Harang
- Department of Mathematics, University of Oslo, P.O. box 1053, Blindern, 0316, Oslo, Norway
| | - Avi Mayorcas
- Centre for Mathematical Sciences, Wilberforce Rd, Cambridge CB3 0WA, UK
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Abstract
AbstractThis 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.
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Lei J. Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces. BERNOULLI 2020. [DOI: 10.3150/19-bej1151] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Dolera E, Regazzini E. Uniform rates of the Glivenko–Cantelli convergence and their use in approximating Bayesian inferences. BERNOULLI 2019. [DOI: 10.3150/18-bej1077] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Weed J, Bach F. Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance. BERNOULLI 2019. [DOI: 10.3150/18-bej1065] [Citation(s) in RCA: 48] [Impact Index Per Article: 9.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Abstract
Abstract
In this paper we study the numerical approximation of the optimal long-run average cost of a continuous-time Markov decision process, with Borel state and action spaces, and with bounded transition and reward rates. Our approach uses a suitable discretization of the state and action spaces to approximate the original control model. The approximation error for the optimal average reward is then bounded by a linear combination of coefficients related to the discretization of the state and action spaces, namely, the Wasserstein distance between an underlying probability measure μ and a measure with finite support, and the Hausdorff distance between the original and the discretized actions sets. When approximating μ with its empirical probability measure we obtain convergence in probability at an exponential rate. An application to a queueing system is presented.
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Dedecker J, Fan X. Deviation inequalities for separately Lipschitz functionals of iterated random functions. Stoch Process Their Appl 2015. [DOI: 10.1016/j.spa.2014.08.001] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Fournier N, Guillin A. On the rate of convergence in Wasserstein distance of the empirical measure. Probab Theory Relat Fields 2014. [DOI: 10.1007/s00440-014-0583-7] [Citation(s) in RCA: 273] [Impact Index Per Article: 27.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Boissard E, Le Gouic T. On the mean speed of convergence of empirical and occupation measures in Wasserstein distance. ANNALES DE L'INSTITUT HENRI POINCARÉ, PROBABILITÉS ET STATISTIQUES 2014. [DOI: 10.1214/12-aihp517] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Maïda M, Maurel-Segala É. Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices. Probab Theory Relat Fields 2013. [DOI: 10.1007/s00440-013-0508-x] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Fathi M, Frikha N. Transport-Entropy inequalities and deviation estimates for stochastic approximation schemes. ELECTRON J PROBAB 2013. [DOI: 10.1214/ejp.v18-2586] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Frikha N, Menozzi S. Concentration bounds for stochastic approximations. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2012. [DOI: 10.1214/ecp.v17-1952] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Boissard E. Simple Bounds for the Convergence of Empirical and Occupation Measures in 1-Wasserstein Distance. ELECTRON J PROBAB 2011. [DOI: 10.1214/ejp.v16-958] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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