One-dimensional empirical measures, order statistics, and Kantorovich transport distances

This work is devoted to the study of rates of convergence of the empirical measures μn = 1 n ∑n k=1 δXk , n ≥ 1, over a sample (Xk)k≥1 of independent identically distributed real-valued random variables towards the common distribution μ in Kantorovich transport distances Wp. The focus is on finite range bounds on the expected Kantorovich distances E(Wp(μn, μ)) or [ E(W p p (μn, μ)) ]1/p in terms of moments and analytic conditions on the measure μ and its distribution function. The study describes a variety of rates, from the standard one 1 √ n to slower rates, and both lower and upperbounds on E(Wp(μn, μ)) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, logconcavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.

[1]  J. Marcinkiewicz.  Sur les fonctions indépendantes , 1938 .

[2]  A. Rényi,et al.  Generalization of an inequality of Kolmogorov , 1955 .

[3]  J. Kiefer,et al.  Asymptotic Minimax Character of the Sample Distribution Function and of the Classical Multinomial Estimator , 1956 .

[4]  Yu. V. Prokhorov.  Convergence of Random Processes and Limit Theorems in Probability Theory , 1956 .

[5]  G. Dall'aglio.  Sugli estremi dei momenti delle funzioni di ripartizione doppia , 1956 .

[6]  A. W. Marshall,et al.  Properties of Probability Distributions with Monotone Hazard Rate , 1963 .

[7]  V. Strassen.  The Existence of Probability Measures with Given Marginals , 1965 .

[8]  R. Dobrushin.  Prescribing a System of Random Variables by Conditional Distributions , 1970 .

[9]  Inequalities for Probabilities of Large Deviations in Terms of Pseudo-Moments , 1971 .

[10]  C. Borell.  Complements of Lyapunov's inequality , 1973 .

[11]  C. Borell.  Convex measures on locally convex spaces , 1974 .

[12]  P. Sen.  Weak Convergence of Multidimensional Empirical Processes for Stationary $\phi$-Mixing Processes , 1974 .

[13]  S. S. Vallender.  Calculation of the Wasserstein Distance Between Probability Distributions on the Line , 1974 .

[14]  Ken-ichi Yoshihara,et al.  Weak convergence of multidimensional empirical processes for strong mixing sequences of stochastic vectors , 1975 .

[15]  E. Lieb,et al.  On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation , 1976 .

[16]  G. Simons,et al.  Inequalities for Ek(X, Y) when the marginals are fixed , 1976 .

[17]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[18]  P. Révész,et al.  Strong Approximations of the Quantile Process , 1978 .

[19]  D. Hensley.  Slicing convex bodies—bounds for slice area in terms of the body’s covariance , 1980 .

[20]  U. Haagerup.  The best constants in the Khintchine inequality , 1981 .

[21]  M. Gromov,et al.  A topological application of the isoperimetric inequality , 1983 .

[22]  A. A. Borovkov,et al.  On an Inequality and a Related Characterization of the Normal Distribution , 1984 .

[23]  János Komlós,et al.  On optimal matchings , 1984, Comb..

[24]  Ludger Riischendorf.  The Wasserstein distance and approximation theorems , 1985 .

[25]  E. Giné,et al.  Empirical Processes Indexed by Lipschitz Functions , 1986 .

[26]  N. Bingham.  EMPIRICAL PROCESSES WITH APPLICATIONS TO STATISTICS (Wiley Series in Probability and Mathematical Statistics) , 1987 .

[27]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[28]  K. Ball.  Logarithmically concave functions and sections of convex sets in $R^{n}$ , 1988 .

[29]  P. Massart.  The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality , 1990 .

[30]  L. Horváth,et al.  Weighted Approximations in Probability and Statistics , 1993 .

[31]  Miklós Simonovits,et al.  Random Walks in a Convex Body and an Improved Volume Algorithm , 1993, Random Struct. Algorithms.

[32]  M. Talagrand,et al.  The Integrability of the Square Exponential Transportation Cost , 1993 .

[33]  Daniel W. Stroock,et al.  Moment estimates derived from Poincar'e and log-arithmic Sobolev inequalities , 1994 .

[34]  Sung K. Ahn,et al.  Strong Approximation of the Quantile Processes and Its Applications under Strong Mixing Properties , 1994 .

[35]  M. Talagrand.  THE TRANSPORTATION COST FROM THE UNIFORM MEASURE TO THE EMPIRICAL MEASURE IN DIMENSION > 3 , 1994 .

[36]  M. Talagrand.  Concentration of measure and isoperimetric inequalities in product spaces , 1994, math/9406212.

[37]  J. Yukich,et al.  Asymptotics for transportation cost in high dimensions , 1995 .

[38]  Sergey G. Bobkov,et al.  Extremal properties of half-spaces for log-concave distributions , 1996 .

[39]  S. Bobkov.  SOME EXTREMAL PROPERTIES OF THE BERNOULLI DISTRIBUTION , 1997 .

[40]  Vladimir M. Zolotarev,et al.  Modern Theory of Summation of Random Variables , 1997 .

[41]  M. Ledoux.  On Talagrand's deviation inequalities for product measures , 1997 .

[42]  C. Houdré,et al.  Isoperimetric constants for product probability measures , 1997 .

[43]  Christian Houdré,et al.  Some Connections Between Isoperimetric and Sobolev-Type Inequalities , 1997 .

[44]  J. Yukich.  Probability theory of classical Euclidean optimization problems , 1998 .

[45]  S. Rachev,et al.  Mass transportation problems , 1998 .

[46]  M. Fradelizi.  Hyperplane Sections of Convex Bodies in Isotropic Position , 1999 .

[47]  On tail probabilities of Kolmogorov-Smirnov statistics based on uniform mixing processes , 1999 .

[48]  S. Bobkov.  Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures , 1999 .

[49]  M. Ledoux.  Concentration of measure and logarithmic Sobolev inequalities , 1999 .

[50]  E. Giné,et al.  Central limit theorems for the wasserstein distance between the empirical and the true distributions , 1999 .

[51]  Olivier Guédon,et al.  Kahane-Khinchine type inequalities for negative exponent , 1999 .

[52]  Maximal inequalities for averages of i.i.d. and 2-exchangeable random variables , 1999 .

[53]  M. Ledoux.  The concentration of measure phenomenon , 2001 .

[54]  A. W. van der Vaart,et al.  Uniform Central Limit Theorems , 2001 .

[55]  Dudley,et al.  Real Analysis and Probability: Measurability: Borel Isomorphism and Analytic Sets , 2002 .

[56]  S. G. Bobkov,et al.  Spectral Gap and Concentration for Some Spherically Symmetric Probability Measures , 2003 .

[57]  C. Villani.  Topics in Optimal Transportation , 2003 .

[58]  M. Ledoux.  Spectral gap, logarithmic Sobolev constant, and geometric bounds , 2004 .

[59]  C. Villani,et al.  Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces , 2005, math/0503123.

[60]  N. Gozlan,et al.  A large deviation approach to some transportation cost inequalities , 2005, math/0510601.

[61]  E. Giné,et al.  Asymptotics for L2 functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances , 2005 .

[62]  L. Kantorovich.  On a Problem of Monge , 2006 .

[63]  L. Kantorovich.  On the Translocation of Masses , 2006 .

[64]  B. Klartag.  A central limit theorem for convex sets , 2006, math/0605014.

[65]  An observation about submatrices , 2008, 0808.2521.

[66]  C. Villani.  Optimal Transport: Old and New , 2008 .

[67]  N. Gozlan.  A characterization of dimension free concentration in terms of transportation inequalities , 2008, 0804.3089.

[68]  Ofer Zeitouni,et al.  An Introduction to Random Matrices: Introduction , 2009 .

[69]  E. Rio.  Upper bounds for minimal distances in the central limit theorem , 2009 .

[70]  A. Guionnet,et al.  An Introduction to Random Matrices , 2009 .

[71]  Christian L'eonard,et al.  Transport Inequalities. A Survey , 2010, 1003.3852.

[72]  S. Bobkov,et al.  Concentration of the information in data with log-concave distributions , 2010, 1012.5457.

[73]  Gaussian concentration for a class of spherically invariant measures , 2010 .

[74]  S. Bobkov,et al.  Concentration of empirical distribution functions with applications to non-i.i.d. models , 2010, 1011.6165.

[75]  S. Bobkov,et al.  On Concentration of Empirical Measures and Convergence to the Semi-circle Law , 2010 .

[76]  M. Meckes,et al.  Concentration and convergence rates for spectral measures of random matrices , 2011, 1109.5997.

[77]  Emmanuel Boissard.  Simple Bounds for the Convergence of Empirical and Occupation Measures in 1-Wasserstein Distance , 2011, 1103.3188.

[78]  S. Dereich,et al.  Constructive quantization: Approximation by empirical measures , 2011, 1108.5346.

[79]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[80]  L. Pastur,et al.  Eigenvalue Distribution of Large Random Matrices , 2011 .

[81]  Sandrine Dallaporta,et al.  EIGENVALUE VARIANCE BOUNDS FOR WIGNER AND COVARIANCE RANDOM MATRICES , 2012, 1203.1597.

[82]  V. Bogachev,et al.  The Monge-Kantorovich problem: achievements, connections, and perspectives , 2012 .

[83]  M. Ledoux,et al.  Analysis and Geometry of Markov Diffusion Operators , 2013 .

[84]  E. Barrio,et al.  The empirical cost of optimal incomplete transportation , 2013, 1310.0924.

[85]  A. Guillin,et al.  On the rate of convergence in Wasserstein distance of the empirical measure , 2013, 1312.2128.

[86]  C. Bordenave,et al.  Combinatorial Optimization Over Two Random Point Sets , 2011, 1103.2734.

[87]  A. Vershik.  Long History of the Monge-Kantorovich Transportation Problem , 2013 .

[88]  E. Barrio,et al.  Rates of convergence for partial mass problems , 2013 .

[89]  Sergey G. Bobkov,et al.  On Concentration Functions of Random Variables , 2015 .

[90]  L. Ambrosio,et al.  A PDE approach to a 2-dimensional matching problem , 2016, 1611.04960.