Sharp Ratio Inequalities for a Conditionally Symmetric Martingale

Volume 58 / 2010

Adam Os/ekowski Bulletin Polish Acad. Sci. Math. 58 (2010), 65-77 MSC: Primary 60G42; Secondary 60G44. DOI: 10.4064/ba58-1-8


Let $f$ be a conditionally symmetric martingale and let $S(f)$ denote its square function.

(i) For $p,\,q>0$, we determine the best constants $C_{p,q}$ such that $$ \sup_n\,{\mathbb E} \frac{|f_n|^p}{(1+S_n^2(f))^q}\leq C_{p,q}. $$ Furthermore, the inequality extends to the case of Hilbert space valued $f$.

(ii) For $N=1,2,\ldots$ and $q>0$, we determine the best constants $C'_{N,q}$ such that $$ \sup_n\,{\mathbb E} \frac{f_n^{2N-1}}{(1+S_n^2(f))^q}\leq C'_{N,q}. $$ These bounds are extended to sums of conditionally symmetric variables which are not necessarily integrable. In addition, we show that neither of the inequalities above holds if the conditional symmetry is not assumed.


  • Adam Os/ekowskiDepartment of Mathematics, Informatics and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland

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