Sharp Ratio Inequalities for a Conditionally Symmetric Martingale
Volume 58 / 2010
Bulletin Polish Acad. Sci. Math. 58 (2010), 65-77
MSC: Primary 60G42; Secondary 60G44.
DOI: 10.4064/ba58-1-8
Abstract
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.
