Exact Estimates for Moments of Random Bilinear Forms

Journal of Theoretical Probability

Volumn 14, Issue 1, 2001, Pages 21-37

  Ibragimov, R ^a   Sharakhmetov, Sh ^b   Cecen, A ^c  

^a YALE UNIVERSITY   (United States)

^b TASHKENT STATE UNIVERSITY OF ECONOMICS   (Uzbekistan)

^c CENTRAL MICHIGAN UNIVERSITY   (United States)

Author keywords

Decoupling; Moment inequalities; Random bilinear forms; Symmetric statistics

Indexed keywords

EID: 0035589874     PISSN: 08949840     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1007812829787     Document Type: Article

References (50)

1. p(G). Ann. Inst. Fourier 20, 335-402.
2. Borovskikh, Yu. V., and Korolyuk, V. S. (1997). Martingale Approximation, VSP, Utrecht, 322 pp.
3. p-radonifying operators. Ann. Probab. 13, 885-897.
4. Cecen, A. A., and Erkal, C. (1996a). Distinguishing between stochastic and deterministic behavior in high frequency foreign exchange rate returns: Can non-linear dynamics help forecasting? International J. Forecast. 12, 465-473.
5. Cecen, A. A., and Erkal, C. (1996b). Distinguishing between stochastic and deterministic behavior in high frequency exchange rate returns: Further evidence. Economics Letters 51, 323-329.
6. De la Peña, V. H. (1992). Decoupling and Khintchine's inequalities for U-statistics. Ann. Probab. 20, 1877-1892.
7. De la Peña, V. H., and Klass, M. J. (1994). Order of magnitude bounds for expectations involving quadratic forms. Ann. Probab. 22, 1044-77.
8. De la Peña, V. H., and Montgomery-Smith (1995). Decoupling inequalities for the tail probabilities of multivariate U-statistics. Ann. Probab. 23, 806-816.
9. Figiel, T., Hitczenko, P., Johnson, W. B., Schechtman, G., and Zinn, J. (1997). Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities. Trans. Amer. Math. Soc. 349, 997-1027.
10. Giné, E., Latała, R., and Zinn, J. (2000). Exponental and moment inequalities for U-statistics. High Dimensional Probability II - Progress in Probability, Birkhäuser, pp. 13-35.
11. Hitczenko, P. (1990). Best constants in martingale version of Rosenthal's inequality. Ann. Probab. 18, 1656-1668.
12. Hitczenko, P. (1994). On a domination of sums of random variables by sums of conditionally independent ones. Ann. Probab. 22, 453-468.
13. Ibragimov, R. (1997). Estimates for the Moments of Symmetric Statistics, Ph.D. Dissertation, Institute of Mathematics of Uzbek Academy of Sciences, Tashkent, 127 pp. (in Russian).
14. Ibragimov, R., and Sharakhmetov, Sh. (1995). On the best constant in Rosenthal's inequality. In Theses of Reports of the Conference on Probability Theory and Mathematical Statistics Dedicated to the 75th Anniversary of Academician S. Kh. Sirajdinov (Fergana, Uzbekistan), Tashkent, pp. 43-44 (in Russian).
15. Ibragimov, R., and Sharahmetov, Sh. (1996a). Bounds for the moments of symmetric statistics. Submitted to "Studia Scientiarum Mathematicarum Hungarica."
16. Ibragimov, R., and Sharakhmetov, Sh. (1996b). Some extremal problems in moment inequalities. To be published in Theory Probab. Appl.
17. Ibragimov, R., and Sharakhmetov, Sh. (1997). On an exact constant for the Rosenthal inequality. Teor. Veroyatnost. i Primen. 42, 341-350 (translation in Theory Probab. Appl. 42 (1997), 294-302 (1998)).
18. Ibragimov, R., and Sharakhmetov, Sh. (1997). On an exact constant for the Rosenthal inequality. Teor. Veroyatnost. i Primen. 42, 341-350 (translation in Theory Probab. Appl. 42 (1997), 294-302 (1998)).
19. Ibragimov, R., and Sharakhmetov, Sh. (1998a). Exact bounds on the moments of symmetric statistics. 7th Vilnius Conference on Probability Theory and Mathematical Statistics. 22nd European Meeting of Statisticians. Abstracts of Communications, Vilnius, Lithuania, pp. 243-244.
20. Ibragimov, R., and Sharakhmetov, Sh. (1998b). The best constant in Rosenthal's inequality for random variables with zero mean. To be published in Theory Probab. Appl.
21. Ibragimov, R., and Sharakhmetov, Sh. (1999). Analogues of Khintchine, Marcinkiewicz-Zygmund and Rosenthal inequalities for symmetric statistics. Scand. J. Statist. 26, 621-633.
22. Ibragimov, R., and Sharakhmetov, Sh. (2000). Moment inequalities for symmetric statistics. In Modern Problems of Probability Theory and Mathematical Statistics. Proceedings of the 4th Fergana International Colloquium on Probability Theory and Mathematical Statistics held 27-29 September, 1995, Tashkent, pp. 184-193 (in Russian); also available at http:// front.math.ucdavis.edu/math.PR/0005004.
23. Johnson, W. B., Schechtman, G., and Zinn, J. (1985). Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. Ann. Probab. 13, 234-253.
24. Khintchine, A. (1923). Uber dyadische Bruche. Math. Z. 18, 109-116.
25. 2-functions of generalized bilinear forms. To appear in Probab. Theory Related Fields.
26. 2-functions of nonnegative random bilinear forms and generalized U-statistics. Ann. Probab. 25, 1471-1501.
27. Koroljuk, V. S., and Borovskich, Yu. V. (1994). Theory of U-statistics, Mathematics and its Applications, Vol. 273, Kluwer Academic Publishers Group, Dordrecht, 552 pp.
28. Krakowiak, W., and Szulga, J. (1986) Random multilinear forms. Ann. Probab. 14, 955-973.
29. Kwapień, S., and Szulga, J. (1991). Hypercontraction methods in moment inequalities for series of independent random variables in normed spaces. Ann. Probab. 19, 1-8.
30. Kwapień, S., and Woyczyński, W. (1992). Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 360 pp.
31. Marcinkiewicz, J., and Zygmund, A. (1937). Sur les fonction independantes. Fund. Math. 29, 60-90.
32. McConnell, T. R., and Taqqu, M. (1986). Decoupling inequalities for multilinear forms in independent symmetric random variables. Ann. Probab. 14, 943-954.
33. Nagaev, S. V. (1990). On a new approach to the study of the distribution of a norm of a random element in Hilbert space. Probability Theory and Mathematical Statistics, Proceedings of the Fifth Vilnius Conference, Mosklas/VSP, Vilnius/Utrecht, pp. 214-226.
34. Nagaev, S. V. (1998). Some refinements of probabilistic and moment inequalities. Theory Probab. Appl. 42, 707-713.
35. Nagaev, S. V., and Pinelis, I. F. (1977). Some inequalities for the distributions of sums of independent random variables. Theory of Probab. Appl. 22, 248-256.
36. Peshkir, G., and Shiryaev, A. N. (1995). Khintchine's inequalities and a martingale extension of the area of their action. Uspekhi Mat. Nauk 50(5), 3-62 (in Russian).
37. Pinelis, I. F. (1980). Estimates for moments of infinite-dimensional martingales. Math. Notes 27, 459-462.
38. 2 test under a symmetry condition. Ann. Probab. 22, 357-368.
39. Pinelis, I. F., and Utev, S. A. (1984). Estimates of moments of sums of independent random variables. Theory Probab. Appl. 29, 574-577.
40. Prokhorov, Yu. V. (1962). Extremal problems in limit theorems. In Proc. VI All-Union Conference on Probability Theory and Mathematical Statistics, Vilnius, pp. 77-84 (in Russian).
41. p (p > 2) spanned by sequences of independent random variables. Israel J. Math. 8, 273-303.
42. Rosiński, J., and Szulga, J. (1982). Product Random Measures and Double Stochastic Integrals, Lecture Notes in Math., Vol. 939, pp. 181-199, Springer, Berlin/New York.
43. Rosiński, J., and Woyczyński, W. A. (1984). Products of Random Measures, Multilinear Forms and Multiple Stochastic Integrals, Lecture Notes in Math., Vol. 1089, pp. 294-315, Springer, Berlin/New York.
44. Rosiński, J., and Woyczyński, W. A. (1986). On Ito stochastic integration with respect to p-stable motion: Inner clock, integrability of sample paths, double and multiple integrals. Ann. Probab. 14, 271-286.
45. Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics, Wiley, New York, 371 pp.
46. Sharakhmetov, Sh. (1995). Estimates for moments of symmetric statistics. In Theses of Reports of the Conference on Probability Theory and Mathematical Statistics Dedicated to the 75th Anniversary of Academician S. Kh. Sirajdinov (Fergana, Uzbekistan), Tashkent, p. 119 (in Russian).
47. Sharakhmetov, Sh. (1997). General Representations for a Joint Distribution of Random Variables and Their Applications, Doctor of Sciences Dissertation. Institute of Mathematics of Uzbek Academy of Sciences, 229 pp. (in Russian).
48. Sjörgen, P. (1982). On the convergence of bilinear and quadratic forms in independent random variables. Studia Math. 71, 285-296.
49. Talagrand (1989). Isoperimetry and integrability of the sum of independent Banach-space valued random variables. Ann. Probab. 17, 1546-1570.
50. Utev, S. A. (1985). Extremal problems in moment inequalities. In Proc. Mathematical Institute of the Siberian Branch of the USSR Academy of Sciences, Vol. 5, pp. 56-75 (in Russian).