On the strong approximation of the non-overlapping k-spacings process with application to the moment convergence rates

[1] S. Alvarez-Andrade and S. Bouzebda, Asymptotic results for hybrids of empirical and partial sums processes. Statist. Papers, 55(4) (2014), 1121–1143. Search in Google Scholar

[2] S. Alvarez-Andrade, S. Bouzebda, On the hybrids of k-spacing empirical and partial sum processes. Rev. Mat. Complut., 30(1) (2017), 185–216. Search in Google Scholar

[3] S. Alvarez-Andrade, S. Bouzebda, Almost sure central limit theorem for the hybrid process. Statistics, 52(3) (2018), 519–532. Search in Google Scholar

[4] S. Alvarez-Andrade, S. Bouzebda, A. Lachal, Some asymptotic results for the integrated empirical process with applications to statistical tests. Comm. Statist. Theory Methods, 46(7) (2017), 3365–3392. Search in Google Scholar

[5] S. Alvarez-Andrade, S. Bouzebda, A. Lachal, Strong approximations for the p-fold integrated empirical process with applications to statistical tests. TEST, 27(4) (2018), 826–849. Search in Google Scholar

[6] E.-E. A. A. Aly, Some limit theorems for uniform and exponential spacings. Canad. J. Statist., 11(3) (1983), 211–219. Search in Google Scholar

[7] E.-E. A. A. Aly, J. Beirlant, L. Horváth, Strong and weak approximations of k-spacings processes. Z. Wahrsch. Verw. Gebiete, 66(3) (1984), 461–484. Search in Google Scholar

[8] Y. Baryshnikov, M. D.Penrose, J. E. Yukich, Gaussian limits for generalized spacings. Ann. Appl. Probab., 19(1) (2009), 158–185. Search in Google Scholar

[9] L. E. Baum, M. Katz, Convergence rates in the law of large numbers. Trans. Amer. Math. Soc., 120 (1965), 108–123. Search in Google Scholar

[10] J. Beirlant, Strong approximations of the empirical and quantile processes of uniform spacings. In Limit theorems in probability and statistics, Vol. I, II (Veszprém, 1982), volume 36 of Colloq. Math. Soc. János Bolyai (1984), pages 77–89. North-Holland, Amsterdam. Search in Google Scholar

[11] J. Beirlant,P.Deheuvels,J.H.J.Einmahl,D.M.Mason, Bahadur-Kiefer theorems for uniform spacings processes. Teor. Veroyatnost. i Primenen., 36(4) (1991), 724–743. Search in Google Scholar

[12] I. Berkes, W. Philipp, Approximation theorems for independent and weakly dependent random vectors. Ann. Probab., 7(1) (1979), 29–54. Search in Google Scholar

[13] P. Billingsley, Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney, (1968). Search in Google Scholar

[14] S. Boucheron, G. Lugosi, P. Massart, Concentration inequalities. Oxford University Press, Oxford. A nonasymptotic theory of independence, With a foreword by Michel Ledoux, (2013). Search in Google Scholar

[15] Y. Chen, T. Li, Moment convergence rates for the uniform empirical process and the uniform sample quantile process. Comm. Statist. Theory Methods, 46(18) (2017), 9086–9091. Search in Google Scholar

[16] M. Csörg˝o, P. Révész, Strong approximations in probability and statistics. Probability and Mathematical Statistics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, (1981). Search in Google Scholar

[17] A. de Acosta, Invariance principles in probability for triangular arrays of B-valued random vectors and some applications. Ann. Probab., 10(2) (1982), 346–373. Search in Google Scholar

[18] P. Deheuvels, Spacings and applications. In Probability and statistical decision theory, Vol. A (Bad Tatzmannsdorf, 1983), pages 1–30. Reidel, Dordrecht, (1985). Search in Google Scholar

[19] P. Deheuvels, Spacings and applications. In Proceedings of 4th Pannonian Symposium on Mathemat. Statist., pages 1–30. Reidel, Dordrecht, (1986). Search in Google Scholar

[20] P. Deheuvels, Non-uniform spacings processes. Stat. Inference Stoch. Process., 14(2) (2011), 141–175. Search in Google Scholar

[21] P. Deheuvels, G. Derzko, Tests of fit based on products of spacings. In Probability, statistics and modelling in public health, pages 119–135. Springer, New York, (2006). Search in Google Scholar

[22] G. E. del Pino, On the asymptotic distribution of k-spacings with applications to goodness-of-fit tests. Ann. Statist., 7(5) (1979), 1058–1065. Search in Google Scholar

[23] J. Durbin, Kolmogorov-Smirnov tests when parameters are estimated with applications to tests of exponentiality and tests on spacings. Biometrika, 62 (1975), 5–22. Search in Google Scholar

[24] M. Ekström, Alternatives to maximum likelihood estimation based on spacings and the Kullback-Leibler divergence. J. Statist. Plann. Inference, 138(6) (2008), 1778–1791. Search in Google Scholar

[25] M. Ekström, Powerful parametric tests based on sum-functions of spacings. Scand. J. Stat., 40(4) (2013), 886–898. Search in Google Scholar

[26] A. Gut, A. Spătaru, Precise asymptotics in the Baum-Katz and Davis laws of large numbers. J. Math. Anal. Appl., 248(1) (2000a), 233–246. Search in Google Scholar

[27] A. Gut, A. Spătaru, Precise asymptotics in the law of the iterated logarithm. Ann. Probab., 28(4) (2000b), 1870–1883. Search in Google Scholar

[28] A. Gut, A. Spătaru, Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables. J. Multivariate Anal., 86(2) (2003), 398–422. Search in Google Scholar

[29] A. Gut, U. Stadtmüller, On the Hsu-Robbins-Erd˝os-Spitzer-Baum-Katz theorem for random fields. J. Math. Anal. Appl., 387(1) (2012), 447–463. Search in Google Scholar

[30] A. Gut, J. Steinebach, Convergence rates in precise asymptotics. J. Math. Anal. Appl., 390(1) (2012), 1–14. Search in Google Scholar

[31] A. Gut, J. Steinebach, Precise asymptotics a general approach. Acta Math. Hungar., 138(4) (2013a), 365–385. Search in Google Scholar

[32] A. Gut, J. Steinebach, Convergence rates in precise asymptotics II. Ann. Univ. Sci. Budapest. Sect. Comput., 39 (2013b), 95–110. Search in Google Scholar

[33] P. L. Hsu, H. Robbins, Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U. S. A., 33 (1947), 25–31. Search in Google Scholar

[34] J. Komlós,P.Major,G.Tusnády, An approximation of partial sums of independent RV’s and the sample DF. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32 (1975), 111–131. Search in Google Scholar

[35] M. Kuo, J. S. Rao, Limit theory and efficiencies for tests based on higher-order spacings. Statistics: applications and new directions (Calcutta, 1981), pages 333–352. Indian Statist. Inst., Calcutta, (1984). Search in Google Scholar

[36] Y.-J. Meng, General laws of precise asymptotics for sums of random variables. J. Korean Math. Soc., 49(4) (2012), 795–804. Search in Google Scholar

[37] J. S. Rao, M. Kuo, Asymptotic results on the Greenwood statistic and some of its generalizations. J. Roy. Statist.Soc.Ser.B, 46(2) (1984), 228–237. Search in Google Scholar

[38] D. M. Mason, W. R.van Zwet, A refinement of the KMT inequality for the uniform empirical process. Ann. Probab., 15(3) (1987), 871–884. Search in Google Scholar

[39] R. Pyke, Spacings. (With discussion.). J. Roy. Statist. Soc. Ser. B, 27 (1965), 395–449. Search in Google Scholar

[40] R. Pyke, Spacings revisited. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. I: Theory of statistics, pages 417–427, Berkeley, Calif. Univ. California Press, (1972). Search in Google Scholar

[41] J. S. Rao, J. Sethuraman, Weak convergence of empirical distribution functions of random variables subject to perturbations and scale factors. Ann. Statist., 3 (1975), 299–313. Search in Google Scholar

[42] J. Sethuraman, J. S. Rao, Pitman efficiencies of tests based on spacings. In Nonparametric Techniques in Statistical Inference (Proc. Sympos., Indiana Univ., Bloomington, Ind., 1969), pages 405–415. Cambridge Univ. Press, London, (1970). Search in Google Scholar

[43] G. R. Shorack, Convergence of quantile and spacings processes with applications. Ann. Math. Statist., 43 (1972), 1400–1411. Search in Google Scholar

[44] R. Singh, N. Misra, U-statistic based on overlapping sample spacings. J. Statist. Plann. Inference, 224 (2023), 98–108. Search in Google Scholar

[45] D. D. Tung,S.R.Jammalamadaka, U-statistics based on higher-order spacings. In Nonparametric statistical methods and related topics, pages 151–169. World Sci. Publ., Hackensack, NJ, (2012) . Search in Google Scholar

[46] D. D. Tung, and S. Rao Jammalamadaka, U-statistics based on spacings. J. Statist. Plann. Inference, 142(3) (2012), 673–684. Search in Google Scholar

[47] D. L. Li, F. X. Zhang, A. Rosalsky, A supplement to the Baum-Katz-Spitzer complete convergence theorem. Acta Math. Sin. (Engl. Ser.), 23(3) (2007), 557–562. Search in Google Scholar

[48] Q. P. Zang, Precise asymptotics for complete moment convergence of U-statistics of i.i.d. random variables. Appl. Math. Lett., 25(2) (2012), 120–127. Search in Google Scholar

[49] Y. Zhang, X.-Y.Yang, Precise asymptotics in the law of the iterated logarithm and the complete convergence for uniform empirical process. Statist. Probab. Lett., 78(9) (2008), 1051–1055. Search in Google Scholar