Weak-type inequality for the martingale square function and a related characterization of Hilbert spaces

• Autorzy: Osękowski A.
• Języki publikacji: EN

Abstrakty

EN

Let f be a martingale taking values in a Banach space B and let S(f) be its square function. We show that if B is a Hilbert space, then P(S(f) ≥1)≤√e∥f∥₁and the constant √e is the best possible. This extends the result of Cox, who established this bound in the real case. Next, we show that this inequality characterizes Hilbert spaces in the following sense: if B is not a Hilbert space, then there is a martingale f for which the above weak-type estimate does not hold.

Słowa kluczowe

EN

martingale; square function; weak type inequality; Banach space; Hilbert space

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2011
• Tom: Vol. 31, Fasc. 2
• Strony: 227--238
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Osękowski A.

• Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

Bibliografia

• [1] D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981), pp. 997-1011.
• [2] D. L. Burkholder, Martingale transforms and geometry of Banach spaces, in: Proceedings of the Third International Conference on Probability in Banach Spaces, Tufts University, 1980, Lecture Notes in Math. Vol. 860 (1981), pp. 35-50.
• [3] D. L. Burkholder, On the number of escapes of a martingale and its geometrical significance, in: Almost Everywhere Convergence, G. A. Edgar and L. Sucheston (Eds.), Academic Press, New York 1989, pp. 159-178.
• [4] D. L. Burkholder, Explorations in martingale theory and its applications, in: École d’Eté de Probabilités de Saint-Flour XIX - 1989, Lecture Notes in Math. Vol. 1464, Springer, Berlin 1991, pp. 1-66.
• [5] D. C. Cox, The best constant in Burkholder’s weak-L1 inequality for the martingale square function, Proc. Amer. Math. Soc. 85 (1982), pp. 427-433.
• [6] J. M. Lee, On Burkholder’s biconvex-function characterization of Hilbert spaces, Proc. Amer. Math. Soc. 118 (2) (1993), pp. 555-559.
• [7] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), pp. 326-350.