Weak type inequality for the square function of a nonnegative submartingale

• Autorzy: Osękowski A.
• Języki publikacji: EN

Abstrakty

EN

Let ƒ be a nonnegative submartingale and S(ƒ) denote its square function. We show that for any λ > 0, λP(S(ƒ) ≥ λ) ≤ π/2||ƒ||1, and the constant π/2 is the best possible. The inequality is strict provided ||ƒ||1 ≠ 0.

Słowa kluczowe

EN

submartingale; square function; weak type inequality

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2009
• Tom: Vol. 57, no 1
• Strony: 81--89
• Opis fizyczny: Bibliogr. 6 poz.

Twórcy

autor

Osękowski A.

• Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland,

Bibliografia

• [1] B. Bollobas, Martingale inequalities, Math. Proc. Cambridge Philos. Soc. 87 (1980), 377-382.
• [2] D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19-42.
• [3] D. L. Burkholder, Martingale and Fourier analysis in Banach spaces, in: Probability and Analysis (Varenna, 1985), Lecture Notes in Math. 1206, Springer, 1986, 61-108.
• [4] D. L. Burkholder, The best constant in the Davis inequality for the expectation of the martingale square function, Trans. Amer. Math. Soc. 354 (2002), 91-105.
• [5] D. Cox, The best constant in Burkholder’s weak-L1 inequality for the martingale square function, Proc. Amer. Math. Soc. 85 (1982), 427-433.
• [6] A. Osękowski, Two inequalities for the first moment of a martingale, its square and maximal function, Bull. Polish Acad. Sci. Math. 53 (2005), 441-449.