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• # Artykuł - szczegóły

## Bulletin of the Polish Academy of Sciences. Mathematics

2010 | 58 | 1 | 65-77

## Sharp Ratio Inequalities for a Conditionally Symmetric Martingale

EN

### Abstrakty

EN
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
$sup_n 𝔼 (|fₙ|^p)/(1+Sₙ²(f))^q ≤ C_{p,q}$.
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
$sup_n 𝔼 (fₙ^{2N-1})(1+Sₙ²(f))^q ≤ C'_{N,q}$.
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.

65-77

wydano
2010

### Twórcy

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