Some sharp inequalities for conditionally symmetric martingales
Wang, Gang
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https://hdl.handle.net/2142/20716
Description
Title
Some sharp inequalities for conditionally symmetric martingales
Author(s)
Wang, Gang
Issue Date
1989
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
Let f be a conditionally symmetric martingale taking values in a Hilbert space $\rm I\!H$ and let S(f) be its square function. If $\nu\sb{\rm p}$ is the smallest positive zero of the confluent hypergeometric function and $\mu\sb{\rm p}$ is the largest positive zero of the parabolic cylinder function of parameter p, then(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\eqalign{\rm\Vert f\Vert\sb{p} \leq \nu\sb{p}\Vert S(f)\Vert\sb{p}\quad&\rm if\quad 0 < p \le 2,\cr\rm\Vert f\Vert\sb{p} \leq \mu\sb{p}\Vert S(f)\Vert\sb{p}\quad&\rm if\quad p \ge 3,\cr\rm\nu\sb{p}\Vert S(f)\Vert\sb{p} \leq \Vert f\Vert\sb{p}\quad&\rm if\quad p \geq 2,\cr}$$(TABLE/EQUATION ENDS)and the above inequalities are sharp.
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