Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2025-01-01T16:19:25.953Z Has data issue: false hasContentIssue false
  • English
  • Français

Some inequalities for martingales and applications to the study of L1

Published online by Cambridge University Press:  24 October 2008

Leonard E. Dor
Leonard E. Dor
Affiliation:
Tel Aviv University, Israel
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain here estimates for the expectation of a restricted quadratic variation of L1 bounded (weak) martingales and positive submartingales, and for related functionals. These inequalities are then used to give a quantitative version of a recent result of D. J. Aldous and D. H. Fremlin concerning the L1 norms of expansions with respect to uniformly integrable L1-normalized martingale difference sequences. Another application relates precisely the degree of norm divergence of an L1-bounded martingale to the degree of non-uniform integrability of that martingale. A partial analogue of this result is proved for certain vector valued martingales with values in L1 which are related to the Radon–Nikodým property.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 1 , January 1981 , pp. 135 - 148
Copyright
Copyright © Cambridge Philosophical Society 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

(1)Aldous, D. J. and Fremlin, D. H.Colacunary sequences in L-spaces. Studia Math. (to appear).Google Scholar
(2)Bourgain, J. and Rosenthal, H. P.Martingales valued in certain subspaces of L 1. Israel J. Math. (to appear).Google Scholar
(3)Burkholder, D. L.Distribution function inequalities for martingales. Ann. Prob. 1 (1973), 19–42.Google Scholar
(4)Burkholder, D. L.Martingale transforms. Ann. Math. Stat. 39 (1968), 1494–1504.Google Scholar
(5)Burkholder, D. L.Maximal inequalities as necessary conditions for almost everywhere convergence. Z. Wahrsch. Verw. Gebiete 3 (1964), 75–88.CrossRefGoogle Scholar
(6)Diestel, J. and Uhl, J. J. Vector Measures. A.M.S. Math. Surveys 15, Providence 1977.Google Scholar
(7)Enflo, P. and Rosenthal, H. P.Some results concerning Lp(μ) spaces. J. Functional Analysis 14 (1973), 325–348.CrossRefGoogle Scholar
(8)Enflo, P. and Starbird, T.Subspaces of L 1 containing L 1. Studia Math. 65 (1979), 203–225.Google Scholar
(9)Gundy, R. F.A decomposition for L 1-bounded martingales. Ann. Math. Stat. 39 (1968), 134–138.Google Scholar
(10)Huff, R. E. and Morris, P. D.Geometric characterizations of the Radon-Nikodým property in Banach spaces. Studia Math. 56 (1976), 157–164.CrossRefGoogle Scholar
(11)Kadec, M. I. and Pelczynski, A.Bases, lacunary sequences and complemented subspaces in the spaces Lp. Studia Math. 21 (1962), 161–176.CrossRefGoogle Scholar
(12)Lewis, D. R. and Stegall, C.Banach spaces whose duals are isomorphic to l 1 (Γ). J. Functional Analysis 12 (1973), 177–181.CrossRefGoogle Scholar
(13)Meyer, P.-A. Martingales and stochastic integrals. I. Springer-Verlag, Lecture Notes in Math. 284, Berlin, Heidelberg, New York 1972.Google Scholar
(14)Nelson, P. I.A class of orthogonal series related to martingales. Ann. Math. Stat. 41 (1970), 1684–1694.Google Scholar
(15)Paley, R. E. A. C.A remarkable series of orthogonal functions. Proc. London Math. Soc. 34 (1932), 241–264.Google Scholar
(16)Rosenthal, H. P.On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37 (1970), 13–36.Google Scholar
(17)Rosenthal, H. P. Seminar talk, University of Illinois 1978.Google Scholar