The rate of Gaussian strong approximation for the sums of i.i.d. multidimensional random vectors

A. Yu. Zaitsev

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 14–2, Tome 364 (2009), pp. 148-165

Résumé

The aim of this paper is to derive new optimal bounds for the rate of strong Gaussian approximation of sums of i.i.d. $\mathbf R^d$-valued random variables $\xi_j$ having finite moments of the form $\mathbf E\,H(|\xi_j|)$, where $H(x)$ is a monotone function growing not slower than $x^{2+\delta}$ and not faster than $e^{cx}$. We obtain some generalizations of the results of U. Einmahl (1989). Bibl. – 44 titles.

Bibliographie

[1] P. Bártfai, “Die Bestimmung der zu einem wiederkehrenden Prozess gehörenden Verteilungfunktion aus den mit Fehlern behafteten Daten einer einzigen Realisation”, Studia Sci. Math. Hungar., 1 (1966), 161–168 | MR | Zbl

[2] I. Berkes, W. Philipp, “Approximation theorems for independent and weakly dependent random vectors”, Ann. Probab., 17 (1979), 29–54 | DOI | MR

[3] E. Berger, Fast sichere Approximation von Partialsummen unabhängiger und stationärer ergodischer Folgen von Zufallsvectoren, Dissertation, Universität Göttingen, 1982

[4] A. A. Borovkov, “O skorosti skhodimosti v printsipe invariantnosti”, Teoriya veroyatn. i ee primen., 18 (1973), 217–234 | MR | Zbl

[5] A. A. Borovkov, A. I. Sakhanenko, “On the rate of convergence in invariance principle”, Lect. Notes Math., 1021, 1981, 59–66 | DOI | MR

[6] A. A. Borovkov, A. I. Sakhanenko, “Skorosti skhodimosti v printsipe invariantnosti dlya banakhovykh prostranstv”, Teoriya veroyatn. i ee primen., 25:4 (1980), 734–744 | MR | Zbl

[7] L. Breiman, “On the tail behaviour of sums of independent random variables”, Z. Wahrscheinlichkeitstheor. verw. Geb., 9 (1967), 20–24 | DOI | MR

[8] M. Csörgő, P. Révész, “A new method to prove Strassen type laws of invariance principle. I”, Z. Wahrscheinlichkeitstheor. verw. Geb., 31 (1975), 255–259 ; “II”, 261–269 | DOI | MR | Zbl | Zbl

[9] M. Csörgő, P. Révész, Strong approximations in probability and statistics, Academic Press, New York, 1981 | MR | Zbl

[10] S. Csörgő, P. Hall, “The Komlós–Major–Tusnády approximations and their applications”, Austral. J. Statist., 26:2 (1984), 189–218 | DOI | MR | Zbl

[11] U. Einmahl, A refinement of the KMT inequality for partial sum strong approximation, Techn. Rep. Ser. Lab. Res. Statist. Probab., 88, Carleton University, University of Ottawa, Ottawa, Canada, 1986

[12] U. Einmahl, “A useful estimate in the multidimensional invariance principle”, Probab. Theor. Rel. Fields, 76 (1987), 81–101 | DOI | MR | Zbl

[13] U. Einmahl, “Strong invariance principles for partial sums of independent random vectors”, Ann. Probab., 15 (1987), 1419–1440 | DOI | MR | Zbl

[14] U. Einmahl, “Extensions of results of Komlós, Major, and Tusnády to the multivariate case”, J. Multivar. Anal., 28 (1989), 20–68 | DOI | MR | Zbl

[15] U. Einmahl, “A new strong invariance principle for sums of independent random vectors”, Zap. nauchn. semin. POMI, 364, POMI, SPb., 2009, 5–31 | MR | Zbl

[16] U. Einmahl, D. M. Mason, “Rates of clustering in Strassen's LIL for partial sums processes”, Probab. Theor. Rel. Fields, 97 (1993), 479–487 | DOI | MR | Zbl

[17] F. Götze, A. Yu. Zaitsev, “Bounds for the rate of strong approximation in the multidimensional invariance principle”, Teoriya veroyatn. i ee primen., 53 (2008), 100–123 | DOI | MR

[18] V. V. Gorodetskii, “O skorosti skhodimosti v mnogomernom printsipe invariantnosti”, Teoriya veroyatn. i ee primen., 20 (1975), 642–649 | Zbl

[19] N. C. Jain, K. Jogdeo, W. F. Stout, “Upper and lower functions for martingales and mixing processes”, Ann. Probab., 3 (1975), 119–145 | DOI | MR | Zbl

[20] J. Komlós, P. Major, G. Tusnády, “An approximation of partial sums of independent RV'-s and the sample DF. I”, Z. Wahrscheinlichkeitstheor. verw. Geb., 32 (1975), 111–131 ; “II”, 34 (1976), 34–58 | DOI | MR | Zbl | DOI | MR | Zbl

[21] P. Major, “The approximation of partial sums of independent r.v.'s”, Z. Wahrscheinlichkeitstheor. verw. Geb., 35 (1976), 213–220 | DOI | MR | Zbl

[22] P. Major, “Approximation of partial sums of i.i.d.r.v.'s when summands have only two moments”, Z. Wahrscheinlichkeitstheor. verw. Geb., 35 (1976), 221–230 | DOI | MR

[23] P. Major, “On the invariance principle for sums of independent identically distributed random variables”, J. Multivar. Anal., 8 (1978), 487–517 | DOI | MR | Zbl

[24] P. Major, “An improvement of Strassen's invariance principle”, Ann. Probab., 7 (1979), 55–61 | DOI | MR | Zbl

[25] V. V. Petrov, Predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Nauka, M., 1987 | MR

[26] W. Philipp, “Almost sure invariance principles for sums of $B$-valued random variables”, Lect. Notes in Math., 709, 1979, 171–193 | DOI | MR | Zbl

[27] Yu. V. Prokhorov, “Skhodimost sluchainykh protsessov i predelnye teoremy teorii veroyatnostei”, Teoriya veroyatn. i ee primen., 1 (1956), 177–238 | Zbl

[28] A. I. Sakhanenko, “Skorost skhodimosti v printsipe invariantnosti dlya raznoraspredelennykh velichin s eksponentsialnymi momentami”, Trudy inst. matem. SO AN SSSR, 3, Nauka, Novosibirsk, 1984, 4–49 | MR

[29] A. I. Sakhanenko, “Otsenki v printsipe invariantnosti”, Trudy inst. matem. SO AN SSSR, 5, Nauka, Novosibirsk, 1985, 27–44 | MR

[30] A. I. Sakhanenko, “O tochnosti silnoi normalnoi approksimatsii v printsipe invariantnosti”, Trudy inst. matem. SO AN SSSR, 13, Nauka, Novosibirsk, 1989, 40–66 | MR

[31] A. I. Sakhanenko, “A new way to obtain estimates in the invariance principle”, High dimensional probability, II (Seattle, 1999), Progr. Probab., 47, Birkhäuser Boston, Boston, 2000, 223–245 | MR | Zbl

[32] A. I. Sakhanenko, “Otsenki v printsipe invariantnosti v terminakh srezannykh stepennykh momentov”, Sibirskii matem. zhurn., 47 (2006), 1355–1371 | MR | Zbl

[33] Qi-Man Shao, “On a problem of Csörgő and Révész”, Ann. Probab., 17 (1989), 809–812 | DOI | MR | Zbl

[34] Qi-Man Shao, “Strong approximation theorems for independent random variables and their applications”, J. Multivar. Anal., 52 (1995), 107–130 | DOI | MR | Zbl

[35] A. V. Skorokhod, Issledovaniya po teorii sluchainykh protsessov, Izd-vo Kievsk. un-ta, Kiev, 1961

[36] V. Strassen, “An invariance principle for the law of iterated logarithm”, Z. Wahrscheinlichkeitstheor. verw. Geb., 3 (1967), 211–226 | DOI | MR

[37] V. Strassen, “Almost sure behavior of sums of independent random variables and martingales”, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, V. II: Contributions to Probability Theory, Part 1 (Berkeley, CA, 1965/66), Univ. California Press, Berkeley, CA, 1967, 315–343 | MR | Zbl

[38] A. Yu. Zaitsev, “Otsenki rasstoyaniya Levi–Prokhorova v mnogomernoi tsentralnoi predelnoi teoreme dlya sluchainykh velichin s konechnymi eksponentsialnymi momentami”, Teoriya veroyatn. i ee primen., 31 (1986), 246–265 | MR

[39] A. Yu. Zaitsev, “Multidimensional version of the results of Komlós, Major, and Tusnády for vectors with finite exponential moments”, ESAIM: Probability and Statistics, 2 (1998), 41–108 | DOI | MR | Zbl

[40] A. Yu. Zaitsev, “Multidimensional version of the results of Sakhanenko in the invariance principle for vectors with finite exponential moments. I”, Teoriya veroyatn. i primen., 45 (2000), 718–738 ; “II”, 46 (2001), 535–561 ; “III”, 46 (2001), 744–769 | DOI | MR | Zbl | MR | Zbl | MR

[41] A. Yu. Zaitsev, “On the strong gaussian approximation in multidimensional case”, Annales de l'I.S.U.P. Publications de l'Institut de Statistique de l'Université de Paris, 45 (2001), 3–7 | MR | Zbl

[42] A. Yu. Zaitsev, “Estimates for the strong approximation in multidimensional Central Limit Theorem”, Proceedings of the International Congress of Mathematicians, Vol. III. Invited Lectures (Bejing, 2002), 2002, 107–116 | MR | Zbl

[43] A. Yu. Zaitsev, “Otsenki tochnosti silnoi approksimatsii v mnogomernom printsipe invariantnosti”, Zap. nauchn. semin. POMI, 339, 2006, 37–53 | MR | Zbl

[44] A. Yu. Zaitsev, “Otsenki tochnosti silnoi gaussovskoi approksimatsii summ nezavisimykh odinakovo raspredelennykh sluchainykh vektorov”, Zap. nauchn. semin. POMI, 351, 2007, 141–157 | MR