Two inequalities for the first moments of a martingale, its square function and its maximal function

• Autorzy: Osękowski A.

Identyfikatory

DOI

10.4064/ba53-4-9

• Języki publikacji: EN

Abstrakty

EN

Given a Hilbert space valued martingale (M_n), let (M^∗_n) and (S_n (M)) denote its maximal function and square function, respectively. We prove that E|M_n |≤ 2ES_n (M), n = 0,1,2,…, EM^∗_n ≤ E|M_n| + 2ES_n (M), n = 0,1,2,…. The first inequality is sharp, and it is strict in all nontrivial cases.

Słowa kluczowe

EN

martingale; square function; maximal function; moment inequality

PL

martyngał

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2005
• Tom: Vol. 53, no 4
• Strony: 441--449
• Opis fizyczny: Bibliogr. 5 poz.

Twórcy

autor

Osękowski A.

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

Bibliografia

• [1] D. Burkholder, Sharp norm comparison of martingale maximal functions and stochastic integrals, in: Proceedings of the Norbert Wiener Centenary Congress (East Lansing, MI, 1994), Proc. Sympos. Appl. Math. 52, Amer. Math. Soc., Providence, RI, 1997, 343-358.
• [2] —, The best constant in the Davis inequality for the expectation of the martingale square function, Trans. Amer. Math. Soc. 354 (2002), 91-105.
• [3] B. Davis, On the integrability of the martingale square function, Israel J. Math. 8 (1970), 187-190.
• [4] A. M. Garsia, The Burgess Davis inequalities via Fefferman’s inequality, Ark. Mat. 11 (1973), 229-237.
• [5] —, Martingale Inequalities: Seminar Notes on Recent Progress, Benjamin, Reading, MA, 1973.