Sharp Ratio Inequalities for a Conditionally Symmetric Martingale

Adam Os/ekowski¹

¹ Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 58 (2010) no. 1, pp. 65-77

Résumé

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.

Détail

DOI : 10.4064/ba58-1-8

Keywords: conditionally symmetric martingale denote its square function determine best constants sup mathbb frac leq furthermore inequality extends hilbert space valued ldots determine best constants sup mathbb frac n leq these bounds extended sums conditionally symmetric variables which necessarily integrable addition neither inequalities above holds conditional symmetry assumed