An Extremal Problem in Uniform Distribution Theory

Baláž, Vladimír; Iacò, Maria Rita; Strauch, Oto; Thonhauser, Stefan; Tichy, Robert F.

doi:10.1515/udt-2016-0012

Abstract

In this paper we consider an optimization problem for Cesàro means of bivariate functions. We apply methods from uniform distribution theory, calculus of variations and ideas from the theory of optimal transport.

References

[1] BALÁŽ, V.—MIŠÍK, L.—STRAUCH, O.—TÓTH, J. T.: Distribution functions of ratio sequences, III, Publ. Math. Debrecen 82 (2013), no. 3-4, 511-529.
Search in Google Scholar Back to article
[2] _____ Distribution functions of ratio sequences, IV, Period. Math. Hungar. 66 (2013) no. 1, 1-22.10.1007/s10998-013-4116-4
Search in Google Scholar Back to article
[3] BEIGLBÖCK, M.—HENRY-LABORDÈRE, P.—PENKNER, F.: Model-independent bounds for option prices—a mass transport approach, Finance Stoch. 17 (2013), no. 3, 477-501.
Search in Google Scholar Back to article
[4] BOSCH, W.: Functions that preserve the uniform distribution of sequences, Trans. Amer. Math. Soc. 307 (1988), no. 1, 143-152.
Search in Google Scholar Back to article
[5| DRMOTA, M.—TICHY, R. F.: Sequences, Discrepancies and Applications, Lecture Notes in Mathematics Vol. 1651, Springer-Verlag, Berlin, 1997.
Search in Google Scholar Back to article
[6] DURANTE, F.—SEMPI, C.: Principles of Copula Theory. CRC/Chapman & Hall, London, 2015.10.1201/b18674
Search in Google Scholar Back to article
[7] FIALOVÁ, J.—STRAUCH, O.: On two-dimensional sequences composed by one-dimensional uniformly distributed sequences, Unif. Distrib. Theory 6 (2011), no. 1, 101–125.
Search in Google Scholar Back to article
[8] GREKOS, G.—STRAUCH, O.: Distribution functions of ratio sequences. II, Unif. Distrib. Theory 2 (2007), no. 1, 53-77.
Search in Google Scholar Back to article
[9] HOFER, M.—IACÒ, M. R.: Optimal bounds for integrals with respect to copulas and applications, Journal of Optimization Theory and Applications 161 (2014), no. 3, 999–1011.
Search in Google Scholar Back to article
[10] IACÒ, M. R.—TICHY, R. F.—THONHAUSER, S.: Distribution functions, extremal limits and optimal transport. Preprint, 2015.10.1016/j.indag.2015.05.003
Search in Google Scholar Back to article
[11] JAWORSKI, P.—DURANTE, F.—HÄRDLE, W.—RYCHLIK, T. (EDS.): Copula Theory and its Applications. In Proc. Lecture Notes in Statistics Vol. 198. Springer, Heidelberg, 2010.10.1007/978-3-642-12465-5
Search in Google Scholar Back to article
[12] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences. In: Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.
Search in Google Scholar Back to article
[13] MCNEIL, A. J.—FREY, R.—EMBRECHTS, P.: Quantitative Risk Management. Princeton Series in Finance. (Concepts, techniques and tools) Princeton University Press, Princeton, NJ, 2005.
Search in Google Scholar Back to article
[14] NELSEN, R. B.: An Introduction to Copulas, second edition. Springer Series in Statistics. Springer, New York, 2000.
Search in Google Scholar Back to article
[15] PILLICHSHAMMER, F.—STEINERBERGER, S.: Average distance between consecutive points of uniformly distributed sequences, Unif. Distrib. Theory 4 (2009), no. 1, 51-67.
Search in Google Scholar Back to article
[16] PORUBSKÝ, Š.—ŠALÁT, T.—STRAUCH, O.: Transformations that preserve uniform distribution, Acta Arith. 49 (1988), no. 5, 459-479.
Search in Google Scholar Back to article
[17] PUCCETTI, G.—ÜSCHENDORF, L. R.: Sharp bounds for sums of dependent risks, J. Appl. Probab. 50 (2013), no. 1, 42-53.
Search in Google Scholar Back to article
[18] ÜSCHENDORE, L. R.: Monge-Kantorovich transportation problem and optimal couplings, Jahresber. Deutsch. Math.-Verein. 109 (2007), no. 3, 113-137.
Search in Google Scholar Back to article
[19] _____ Mathematical Risk Analysis. Springer Series in Operations Research and Financial Engineering. (Dependence, Risk Bounds, Optimal Allocations and Portfolios). Springer, Heidelberg, 2013.
Search in Google Scholar Back to article
[20] ÜSCHENDORE, L. R.—UCKELMANN, L.: Numerical and analytical results for the transportation problem of Monge-Kantorovich, Metrika 51 (2000), no. 3, 245-258 (electronic).
Search in Google Scholar Back to article
[21] STEINERBERGER, S.: Uniform distribution preserving mappings and variational problems, Unif. Distrib. Theory 4 (2009), no. 1, 117-145.
Search in Google Scholar Back to article
[22] STRAUCH, O.—PORUBSKÝ, Š.: Distribution of Sequences: a Sampler, Schriftenreihe der Slowakischen Akademie der Wissenschaften Vol. 1, Series of the Slovak Academy of Sciences. Peter Lang, Frankfurt am Main, 2005.
Search in Google Scholar Back to article
[23] STRAUCH, O.—TÓTH, J. T.: Distribution functions of ratio sequences, Publ. Math. Debrecen 58 (2001), no. 4, 751-778.
Search in Google Scholar Back to article
[24] TICHY, R. F.—WINKLER, R.: Uniform distribution preserving mappings, Acta Arith. 60 (1991), no. 2. 177-18.
Search in Google Scholar Back to article
[25] UCKELMANN, L.: Optimal couplings between one-dimensional distributions. In: Distributions with given marginals and moment problems (Prague, 1996), Kluwer Acad. Publ., Dordrecht, 1997, pp. 275-281.10.1007/978-94-011-5532-8_32
Search in Google Scholar Back to article
[26] VAN DER. CORPUT, J. G.: Verteilungsfunktionen I-II. Proc. Akad. Amsterdam 38 (1935), 813-821, 1058-1066.
Search in Google Scholar Back to article
[27] _____ Verteilungsfunktionen III-VIII, Proc. Akad. Amsterdam 39 (1936), 10-19, 19-26, 149-153, 339-344, 489-494, 579-590.
Search in Google Scholar Back to article
[28] VILLANI, C.: Optimal Transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Vol. 338. Springer-Verlag, Berlin, 2009.10.1007/978-3-540-71050-9
Search in Google Scholar Back to article
[29] WEINSTOCK, R.: Calculus of Variations. Dover Publications, Inc., New York, 1974. (With applications to Physics and Engineering, Reprint of the 1952 edition.)
Search in Google Scholar Back to article

An Extremal Problem in Uniform Distribution Theory

Abstract

Paradigm

My account