Operators on martingales, ф-summing operators, and the contraction principle

• Autorzy: Geiss S.
• Języki publikacji: EN

Abstrakty

EN

For the absolutely Ф-summmg operators T: X→Y between Banach spaces X and Y we consider martingale inequalities of the type…[formula] where ..[formula]…is a martingale difference sequence and i is a sequence of normalized functionals on X, and we show that these inequalities are useful in different directions. For example, for a Banach space X, x₁…x_n ∈X, independent standard Gaussian variables g_n, and 1 < r < ∞ we deduce that..[formula]… where is a scale-valued martingale difference sequence such that [formula]…is predictable ..[formula].. is a sequence of stopping times and [formula].

Słowa kluczowe

EN

absolutely Φ-summing operators; martingale ineqaality; contraction principle; Rademacher variables

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 1998
• Tom: Vol. 18, Fasc. 1
• Strony: 149--171
• Opis fizyczny: Bibliogr. 32 poz.

Twórcy

autor

Geiss S.

• Mathematisches Institut der Friedrich-Schiller-Universität Jena, Postfach, D-07740 Jena, Germany

Bibliografia

• [1] P. Assouad, Applications sommantes et radonifiantes, Ann. Inst. Fourier 22 (3) (1972), pp. 81-93.
• [2] N. L. Bassily and J. Mogyoródi, On the ВМОф-spaces with general Young function, Ann. Univ. Sci. Budapest Eötvös, Sect. Math. 27 (1984), pp. 215-227.
• [3] C. Bennet and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
• [4] J. Вourgain, On the behaviour of the constant in the Littlewood-Paley inequality, in: Israel Seminar, GAFA 1987/88, J. Lindenstrauss and V. D. Milman (Eds.), Lecture Notes in Math. 1376, Springer, 1989, pp. 202-208.
• [5] D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), pp. 19-42.
• [6] — Boundary value problems and sharp inequalities for martingale transforms, ibidem 12 (1984), pp. 647-702.
• [7] — Explorations in martingale theory and its applications, in: Ecole d’Eté de Probabilités de Saint-Flour XIX-1989, Lecture Notes in Math. 1464, Springer, 1991, pp. 1-66.
• [8] — B. J. Davis and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, in: Proceedings of the Sixth Berkeley Symposium, Math. Statist. Probab. 2 (1972), pp. 223-240.
• [9] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasilinear operators on martingales, Acta Math. 124 (1970), pp. 249-304.
• [10] S. Chang, M. Wilson and J. Wolff, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv. 60 (1985), pp. 217-246.
• [11] S. Chevet, Series de variables aléatoires gaussiens á valeurs dans E®eF. Applications aux produits d’espaces de Wiener abstraits, Séminaire Maurey-Schwartz, Exposé XIX, 1977/78.
• [12] B. Davis, On the integrability of the martingale square function, Israel J. Math. 8 (1970), pp. 187-190.
• [13] S. Geiss, ВМОф-spaces and applications to extrapolation theory, Studia Math. 122 (3) (1997), pp. 235-274.
• [14] — and M. Junge, Type and cotype with respect to arbitrary orthonormal systems, J. Approx. Theory 82 (3) (1995), pp. 399-433.
• [15] P. Hitczenko, Upper bound for the Lp-norms of martingales, Probab. Theory Related Fields 86 (1990), pp. 225-238.
• [16] — Domination inequality for martingale transforms of a Rademacher sequence, Israel J. Math. 84 (1993), pp. 161-178.
• [17] J. Hoffmann-J0rgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), pp. 159-186.
• [18] — Probability in Banach spaces, in: Ecole d'Eté de Probabilités de Saint-Flour VI-1976, Lecture Notes in Math. 598, Springer, 1977, pp. 1-186.
• [19] R. C. James, Super-reflexive Banach spaces, Canad. J. Math. 5 (1972), pp. 896-904.
• [20] — Non-reflexive spaces of type 2, Israel J. Math. 30 (1978), pp. 1-13.
• [21] J.-P. Kahane, Some random series of functions, in: Heath Mathematical Monographs, 2nd edition, Cambridge Univ. Press, 1985.
• [22] S. Kwapień and W. A, Woyczyński, Random series and stochastic integrals: Single and multiple, in: Probability and Its Applications, Birkhäuser, Boston-Basel-Berlin 1992.
• [23] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, 1991.
• [24] W. Linde and A. Pietsch, Mappings of Gaussian cylindrical measures in Banach spaces, Theory Probab. Appl. 19 (1974), pp. 445-460.
• [25] R. Sh. Liptser and A. N. Shirуayev, Theory of Martingales, Kluwer Academic Publishers, Dordrecht-Boston-London 1989.
• [26] B. Maure y et G. Pisíer, Series de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), pp. 45-90.
• [27] A. Pietsch, Operator Ideals, North-Holland, Amsterdam 1980.
• [28] — and J. Wenzel, Orthonormal Systems and Banach Space Geometry, Cambridge Univ. Press (to appear).
• [29] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), pp. 326-350.
• [30] M. Talagrand, Regularity of Gaussian processes, Acta Math. 159 (1987), pp. 99-149.
• [31] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-dimensional Operator Ideals, Pitman, New York 1989.
• [32] G. Wang, Some Sharp Inequalities for Conditionally Symmetric Martingales, PhD Thesis, University of Illinois, 1989.