Sharp Ratio Inequalities for a Conditionally Symmetric Martingale

Adam Osękowski

Adam Osękowski

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 58 (2010), p. 65-77 / Harvested from The Polish Digital Mathematics Library

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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 $s u p_{n} {(| f ₙ |}^{p} {) / (1 + S ₙ ² (f))}^{q} \leq C_{p, q}$ . Furthermore, the inequality extends to the case of Hilbert space valued f. (ii) For N = 1,2,... and q > 0, we determine the best constants $C_{N, q}^{'}$ such that $s u p_{n} (f ₙ^{2 N - 1}) {(1 + S ₙ ² (f))}^{q} \leq C_{N, q}^{'}$ . 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.

Publié le : 2010-01-01

Zbl 1193.60058

EUDML-ID : urn:eudml:doc:281210

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     title = {Sharp Ratio Inequalities for a Conditionally Symmetric Martingale},
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     year = {2010},
     pages = {65-77},
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Adam Osękowski. Sharp Ratio Inequalities for a Conditionally Symmetric Martingale. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 58 (2010) pp. 65-77. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba58-1-8/