Rosenthal type inequalities for martingales in symmetric spaces

S. V. Astashkin

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2012), pp. 60-66.

Résumé

Inequalities similar to the classical Rosenthal inequalities are proved for sequences of martingale differences in general symmetric spaces. A central role is played by the predictable square variation of a martingale.

Détail

Mots-clés : martingale
Keywords: martingale differences, Rosenthal inequality, symmetric space, Boyd indices.

Bibliographie

[1] Rosenthal H. P., “On the subspaces of $L_p$ $(p>2)$ spanned by sequences of independent random variables”, Israel J. Math., 8 (1970), 273–303 | DOI | MR | Zbl

[2] Johnson W. B., Schechtman G., “Sums of independent random variables in rearrangement invariant function spaces”, Ann. Probab., 17:2 (1989), 789–808 | DOI | MR | Zbl

[3] Johnson W. B., Maurey B., Schechtman G., Tzafriri L., Symmetric structures in Banach spaces, Mem. Amer. Math. Soc., 1, no. 217, 1979 | MR

[4] Lindenstrauss J., Tzafriri L., Classical Banach spaces, v. II, Springer-Verlag, Berlin, 1979 | MR | Zbl

[5] Astashkin S. V., Sukochev F. A., “Sums of independent random variables in symmetric spaces: an operator approach”, Israel J. Math., 145 (2005), 125–156 | DOI | MR | Zbl

[6] Astashkin S. V., Sukochev F. A., “Ryady nezavisimykh funktsii s nulevym srednim v simmetrichnykh prostranstvakh so svoistvom Kruglova”, Issl. po lin. oper. i teor. funktsii, Zap. nauchn. semin. POMI, 345, 2007, 25–50 | MR

[7] Astashkin S. V., Sukochev F. A., “Nezavisimye funktsii i geometriya banakhovykh prostranstv”, UMN, 65:6 (2010), 3–86 | DOI | MR

[8] Braverman M. Sh., Independent random variables and symmetric spaces, Cambridge University Press, Melbourne, 1994

[9] Kruglov V. M., “Zamechanie k teorii bezgranichno delimykh zakonov”, Teoriya veroyatn. i ee primen., 15:2 (1970), 330–336 | MR | Zbl

[10] Burkholder D. L., Gandy R. F., “Extrapolation and interpolation of quasi-linear operators on martingales”, Acta Math., 124:1 (1970), 249–304 | DOI | MR | Zbl

[11] Burkholder D. L., Davis B. J., Gandy R. F., “Integral inequalities for convex functions of operators on martingales”, Proc. of Sixth Berkeley Symp. Math. Statist. Prob., v. II, Probability theory, Univ. California Press, Berkeley, CA, 1972, 223–240 | MR | Zbl

[12] Burkholder D. L., “Distribution function inequalities for martingales”, Ann. Probab., 1:1 (1973), 19–42 | DOI | MR | Zbl

[13] Peshkir G., Shiryaev A. N., “Neravenstva Khinchina i martingalnoe rasshirenie sfery ikh deistviya”, UMN, 50:5 (1995), 3–62 | MR | Zbl

[14] Astashkin S. V., Sukochev F. A., Wong C. P., “Disjointification of martingale differences and conditionally independent random variables with some applications”, Studia Math., 205:2 (2011), 171–200 | DOI | MR | Zbl

[15] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR

[16] Kashin B. S., Saakyan A. A., Ortogonalnye ryady, AFTs, M., 1999 | MR | Zbl

[17] Shiryaev A. N., Veroyatnost, Nauka, M., 1980 | MR | Zbl

[18] Elliott R., Stokhasticheskii analiz i ego prilozheniya, Mir, M., 1986 | MR | Zbl

[19] Neveu J., Discrete parameter martingales, Amsterdam, 1975 North–Holland | MR | Zbl

[20] Hitczenko P., Montgomery-Smith S., “Measuring the magnitude of sums of independent random variables”, Ann. Probab., 29:1 (2001), 447–466 | DOI | MR | Zbl