Sharp Ratio Inequalities for a Conditionally Symmetric Martingale

• Autorzy: Adam Osękowski
• Języki publikacji: 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.

Kategorie tematyczne

• 60G42: Martingales with discrete parameter
• 60G44: Martingales with continuous parameter

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2010
• Tom: 58
• Numer: 1
• Strony: 65-77

Daty

wydano

2010

Twórcy

autor

Adam Osękowski

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

Identyfikatory

DOI

10.4064/ba58-1-8