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Zap. Nauchn. Sem. POMI, 2009, Volume 364, Pages 148–165 (Mi znsl3155)

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

References

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  mathscinet  zmath
2. I. Berkes, W. Philipp, “Approximation theorems for independent and weakly dependent random vectors”, Ann. Probab., 17 (1979), 29–54  crossref  mathscinet
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  mathnet  mathscinet  zmath
5. A. A. Borovkov, A. I. Sakhanenko, “On the rate of convergence in invariance principle”, Lect. Notes Math., 1021, 1981, 59–66  crossref  mathscinet
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  mathnet  mathscinet  zmath
7. L. Breiman, “On the tail behaviour of sums of independent random variables”, Z. Wahrscheinlichkeitstheor. verw. Geb., 9 (1967), 20–24  crossref  mathscinet
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  crossref  mathscinet  zmath; “II”, 261–269  zmath
9. M. Csörgő, P. Révész, Strong approximations in probability and statistics, Academic Press, New York, 1981  mathscinet  zmath
10. S. Csörgő, P. Hall, “The Komlós–Major–Tusnády approximations and their applications”, Austral. J. Statist., 26:2 (1984), 189–218  crossref  mathscinet  zmath
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  crossref  mathscinet  zmath  isi
13. U. Einmahl, “Strong invariance principles for partial sums of independent random vectors”, Ann. Probab., 15 (1987), 1419–1440  crossref  mathscinet  zmath  isi
14. U. Einmahl, “Extensions of results of Komlós, Major, and Tusnády to the multivariate case”, J. Multivar. Anal., 28 (1989), 20–68  crossref  mathscinet  zmath  isi
15. U. Einmahl, “A new strong invariance principle for sums of independent random vectors”, Zap. nauchn. semin. POMI, 364, POMI, SPb., 2009, 5–31  mathnet  mathscinet  zmath
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  crossref  mathscinet  zmath  isi
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  mathnet  crossref  mathscinet  elib
18. V. V. Gorodetskii, “O skorosti skhodimosti v mnogomernom printsipe invariantnosti”, Teoriya veroyatn. i ee primen., 20 (1975), 642–649  mathnet  zmath
19. N. C. Jain, K. Jogdeo, W. F. Stout, “Upper and lower functions for martingales and mixing processes”, Ann. Probab., 3 (1975), 119–145  crossref  mathscinet  zmath
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  crossref  mathscinet  zmath; “II”, 34 (1976), 34–58  crossref  mathscinet  zmath
21. P. Major, “The approximation of partial sums of independent r.v.'s”, Z. Wahrscheinlichkeitstheor. verw. Geb., 35 (1976), 213–220  crossref  mathscinet  zmath
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  crossref  mathscinet
23. P. Major, “On the invariance principle for sums of independent identically distributed random variables”, J. Multivar. Anal., 8 (1978), 487–517  crossref  mathscinet  zmath
24. P. Major, “An improvement of Strassen's invariance principle”, Ann. Probab., 7 (1979), 55–61  crossref  mathscinet  zmath
25. V. V. Petrov, Predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Nauka, M., 1987  mathscinet
26. W. Philipp, “Almost sure invariance principles for sums of $B$-valued random variables”, Lect. Notes in Math., 709, 1979, 171–193  crossref  mathscinet  zmath
27. Yu. V. Prokhorov, “Skhodimost sluchainykh protsessov i predelnye teoremy teorii veroyatnostei”, Teoriya veroyatn. i ee primen., 1 (1956), 177–238  mathnet  zmath
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  mathscinet
29. A. I. Sakhanenko, “Otsenki v printsipe invariantnosti”, Trudy inst. matem. SO AN SSSR, 5, Nauka, Novosibirsk, 1985, 27–44  mathscinet
30. A. I. Sakhanenko, “O tochnosti silnoi normalnoi approksimatsii v printsipe invariantnosti”, Trudy inst. matem. SO AN SSSR, 13, Nauka, Novosibirsk, 1989, 40–66  mathscinet
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  mathscinet  zmath
32. A. I. Sakhanenko, “Otsenki v printsipe invariantnosti v terminakh srezannykh stepennykh momentov”, Sibirskii matem. zhurn., 47 (2006), 1355–1371  mathnet  mathscinet  zmath  elib
33. Qi-Man Shao, “On a problem of Csörgő and Révész”, Ann. Probab., 17 (1989), 809–812  crossref  mathscinet  zmath  isi
34. Qi-Man Shao, “Strong approximation theorems for independent random variables and their applications”, J. Multivar. Anal., 52 (1995), 107–130  crossref  mathscinet  zmath  isi
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  crossref  mathscinet
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  mathscinet  zmath
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  mathnet  mathscinet
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  crossref  mathscinet  zmath
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  mathnet  crossref  mathscinet  zmath; “II”, 46 (2001), 535–561  mathnet  mathscinet  zmath; “III”, 46 (2001), 744–769  mathnet  mathscinet
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  mathscinet  zmath
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  mathscinet  zmath
43. A. Yu. Zaitsev, “Otsenki tochnosti silnoi approksimatsii v mnogomernom printsipe invariantnosti”, Zap. nauchn. semin. POMI, 339, 2006, 37–53  mathnet  mathscinet  zmath
44. A. Yu. Zaitsev, “Otsenki tochnosti silnoi gaussovskoi approksimatsii summ nezavisimykh odinakovo raspredelennykh sluchainykh vektorov”, Zap. nauchn. semin. POMI, 351, 2007, 141–157  mathnet  mathscinet


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