Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 9 (2012) open journal systems 

Quantile coupling inequalities and their applications

David M. Mason, University of Delaware
Harrison Zhou, Yale University

This is partly an expository paper. We prove and highlight a quantile inequality that is implicit in the fundamental paper by Komlós, Major, and Tusnády [31] on Brownian motion strong approximations to partial sums of independent and identically distributed random variables. We also derive a number of refinements of this inequality, which hold when more assumptions are added. A number of examples are detailed that will likely be of separate interest. We especially call attention to applications to the asymptotic equivalence theory of

AMS 2000 subject classifications: Primary 62E17; secondary 62B15, 62G05.

Keywords: Quantile coupling, large deviation, KMT construction, asymptotic equivalence, function estimation.

Creative Common LOGO

Full Text: PDF

Mason, David M., Zhou, Harrison, Quantile coupling inequalities and their applications, Probability Surveys, 9, (2012), 439-479 (electronic). DOI: 10.1214/12-PS198.


[1]    Amosova, N. N. (1999). The necessity of the Linnik condition in a theorem on probabilities of large deviations. Journal of Mathematical Sciences 93 255–258. MR1449842

[2]    Arak, T. V. and Zaitsev, A. Yu. (1988). Uniform Limit Theorems for Sums of Independent Random Variables. Translation of Trudy. Mat. Inst. Steklov 174 1986. Proc. Steklov Inst. Math. MR0871856

[3]    Bártfai, P. (1966). Die Bestimmung der zu einem wiederkehrenden Prozess gehörenden Verteilungsfunktion aus den mit Fehlern behafteten Daten einer einzigen Realisation. Studia Sci. Math. Hungar 1 161–168. MR0215377

[4]    Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29–54. MR0515811

[5]    Bolthausen, E. (1982). Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 672–688. MR0659537

[6]    Bretagnolle, J. and Massart, P. (1989). Hungarian constructions from the nonasymptotic view point. Ann. Probab. 17 239–256. MR0972783

[7]    Brown, L.D., Carter, A.V., Low, M.G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 2074–2097. MR2102503

[8]    Cai, T. T. and Zhou, H. H. (2009). Asymptotic Equivalence and Adaptive Estimation for Robust Nonparametric Regression. Ann. Statist. 37 3204–3235. MR2549558

[9]    Carter, A. V. and Pollard, D. (2004). Tusnády’s inequality revisited. Ann. Statist., 32 2731–2741. MR2154001

[10]    Chatterjee, S. (2012). A new approach to strong embeddings. Probability Theory and Related Fields 152 231–264. MR2875758

[11]    Csörgʺo  , M. and Horváth, L. (1993). Weighted Approximations in Probability and Statistics, John Wiley & Sons, Chichester etc. MR1215046

[12]    Csörgʺo   , M. and Révész, P. (1981). Strong Approximations in Probability and Statistics, Academic, New York. MR0666546

[13]    Csörgʺo   , M., Csörgoʺ   , S., Horváth, L. and Mason, D. M. (1986). Weighted empirical and quantile processes. Ann. Probab. 14 31–85. MR0815960

[14]    de la Peña, V. H., Lai, T. L. and Shao, Q-M. (2009). Self-normalized Processes. Limit Theory and Statistical Applications. Probability and its Applications (New York). Springer-Verlag, Berlin. MR2488094

[15]    Donoho, D. L. and Johnstone, I. M. (1995). Adapt to unknown smoothness via wavelet shrinkage. J. Amer. Stat. Assoc. 90 1200–1224. MR1379464

[16]    Donoho, D. L. and Yu , T. P.-Y. (2000). Nonlinear Pyramid Transforms Based on Median-Interpolation. SIAM Journal of Math. Anal. 31 1030–1061. MR1759198

[17]    Dudley, R. M. (2000). An exposition of Bretagnolle and Massart’s proof of the KMT theorem for the uniform empirical process. In: Notes on empirical processes, lectures notes for a course given at Aarhus University, Denmark, August 2000.

[18]    Einmahl, U. (1986). A refinement of the KMT-inequality for partial sum strong approximation. Technical Report Series of the Laboratory for Research in Statistics and Probability, Carleton University-University of Ottawa, No. 88.

[19]    Einmahl, U. (1989). Extensions of results of Komlós, Major, and Tusnády to the multivariate case. J. Multivariate Anal. 28 20–68. MR0996984

[20]    Giné, E., Götze, F. and Mason, D.M. (1997). When is the Student t-statistic asymptotically standard normal? Ann. Probab. 25 1514–1531. MR1457629

[21]    Golubev, G. K., Nussbaum, M. and Zhou, H. H. (2010). Asymptotic equivalence of spectral density estimation and Gaussian white noise. Ann. Statist. 38 181–214. MR2589320

[22]    Grama, I. and Haeusler, E. (2000). Large deviations for martingales via Cramér’s method. Stochastic Process. Appl. 85 279–293. MR1731027

[23]    Grama, I. and Nussbaum, M. (1998). Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Relat. Fields 111 167–214 MR1633574

[24]    Grama, I. and Nussbaum, M. (2002a). Asymptotic equivalence for nonparametric regression. Mathematical Methods of Statistics, 11 11–36. MR1900972

[25]    Grama, I. and Nussbaum, M. (2002b). A functional Hungarian construction for sums of independent random variables. En l’honneur de J. Bretagnolle, D. Dacunha-Castelle, I. Ibragimov. Ann. Inst. H. Poincaré Probab. Statist. 38 923–957. MR1955345

[26]    Hall, P. and Patil, P. (1996). On the choice of smoothing parameter, threshold and truncation in nonparametric regression by wavelet methods, J. Roy. Statist. Soc. Ser. B, 58 361–377. MR1377838

[27]    Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30. MR0144363

[28]    Hu, Z., Robinson, J. and Wang, Q. (2007). Cramér-type large deviations for samples from a finite population. Ann. Statist. 35 673–696. MR2336863

[29]    Jing, B.-Y, Shao, Q-M. and Wang, Q. (2003). Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab. 31 2167–2215. MR2016616

[30]    Koltchinskii, V. (1994). Komlós-Major-Tusnády approximation for the general empirical process and Haar expansions of classes of functions. J. Theoret. Probab. 7 73–118. MR1256392

[31]    Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent rv’s and the sample df. I Z. Wahrsch. verw. Gebiete 32 111–131. MR0375412

[32]    Komlós, J., Major, P. and Tusnády, G. (1976). An approximation of partial sums of independent rv’s and the sample df. II Z. Wahrsch. verw. Gebiete 34 33–58. MR0402883

[33]    Lawler, G, F. and Trujillo Ferreras, J. A. (2007). Random walk loop soup. Trans. Amer. Math. Soc. 359 767–787. MR2255196

[34]    Major, P. (1976). The approximation of partial sums of independent RV’s. Z. Wahrsch. verw. Gebiete 35 213–220. MR0415743

[35]    Major, P. (1978). On the invariance principle for sums of independent identically distributed random variables. J. Multivariate Anal. 8 487–517. MR0520959

[36]    Major, P. (1999). The approximation of the normalized empirical ditribution function by a Brownian bridge. Technical report, Mathematical Institute of the Hungarian Academy of Sciences. Notes available from http://www.renyi.hu/\textasciitildemajor/probability/empir.html.

[37]    Mason, D. M. (2001). Notes on the KMT Brownian bridge approximation to the uniform empirical process. In Asymptotic Methods in Probability and Statistics with Applications (N. Balakrishnan, I. A. Ibragimov and V. B. Nevzorov, eds.) 351–369. Birkhäuser, Boston. MR1890338

[38]    Mason, D. M. and van Zwet, W. R. (1987). A refinement of the KMT inequality for the uniform empirical process. Ann. Probab. 15 871–884 MR0893903

[39]    Massart, P. (1990). The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 1269–1283. MR1062069

[40]    Massart, P. (2002). Tusnády’s lemma, 24 years later. Ann. Inst. H. Poincaré Probab. Statist. 38 991–1007. MR1955348

[41]    Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399–2430 MR1425959

[42]    Petrov, V. V. (1975). Sums of Independent Random Variables. Springer-Verlag. (English translation from 1972 Russian edition). MR0388499

[43]    Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press Oxford 1995 MR1353441

[44]    Richter, H. Über das dyadische Einbettungsschema von Komlós-Major-Tusnády. Diplomarbeit, Ruhr-Universitaet Bochum, 1978; Referent: P. Gänssler, 1978

[45]    Sakhanenko, A. I. (1982). On unimprovable estimates of the rate of convergence in invariance principle. Colloquia Math. Soc. János Bolyai, 32 II. Nonparametric Statistical Inference, 779–783. Ed. by Gdnenko, B.V., Vincze, I. and Puri, M.L; North Holland, Amsterdam. MR0669045

[46]    Sakhanenko, A. I. (1985a). Convergence rate in the invariance principle for non-identically distributed variables with exponential moments. In: Advances in Probability Theory: Limit Theorems for Sums of Random Variables (A. A. Borovkov, ed.) 2–73. Springer, New York.

[47]    Sakhanenko, A. I. (1985b). Estimates in the invariance principles, In: Trudy Inst. Mat. SO AN SSSR, vol. 5, Nauka, Novosibirsk, pp 27-44. (Russian) MR0821751

[48]    Sakhanenko, A. I. (1991). On the accuracy of normal approximation in the invariance principle Siberian Adv. Math. 1 58–91. MR1138005

[49]    Sakhanenko, A. I. (1996). Estimates for the Accuracy of Coupling in the Central Limit Theorem. Siberian Mathematical Journal 37 811–823. MR1643327

[50]    Saulis, L. I. (1969). Asymptotic expansions of probabilities of large deviations (in Russian). Litovskii Matemat. Sbomik (Lietuvos. Matematikos Rinkinys) 9 605–625. MR0264742

[51]    Saulis, L. I. and Statulevicius, V.A. (1991). Limit theorems for large deviations. Kluwer Academic Publishers. MR1171883

[52]    Shao, Q-M. (1995). Strong approximation theorems for independent random variables and their applications. J. Multivariate Anal. 52 107-130. MR1325373

[53]    Shorack, G. R. and Wellner, J. A. (1986). Empirical processes with applications to statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York. MR0838963

[54]    Statulevičius, V. and Jakimavičius, D. (1988). Estimates of semi-invariants and centered moments of stochastic processes with mixing. I. Lithuanian Mathematical Journal (Translated from Lietuvos Matematikos Rinkinys) 28 112–129. MR0949647

[55]    Tusnády, G. (1977). A Study of Statistical Hypotheses. Dissertation, The Hungarian Academy of Sciences, Budapest. (In Hungarian.)

[56]    Wolf, W. (1977). Asymptotische Entwicklungen für Wahrscheinlichkeiten grosser Abweichungen. Z. Wahrsch. verw. Geb. 40 239–256. MR0455089

[57]    Zhou, H. H. (2004). Minimax Estimation with Thresholding and Asymptotic Equivalence for Gaussian Variance Regression. Ph.D. Dissertation. Cornell University, Ithaca, NY. MR2706282

Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787