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On the behavior of the constant in a decoupling inequality for martingales

Author: Paweł Hitczenko
Journal: Proc. Amer. Math. Soc. 121 (1994), 253-258
MSC: Primary 60G42
MathSciNet review: 1176481
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Abstract: Let $ ({f_n})$ and $ ({g_n})$ be two martingales with respect to the same filtration $ ({\mathcal{F}_n})$ such that their difference sequences $ ({d_n})$ and $ ({e_n})$ satisfy

$\displaystyle P({d_n} \geq \lambda \vert{\mathcal{F}_{n - 1}}) = P({e_n} \geq \lambda \vert{\mathcal{F}_{n - 1}})$

for all real $ \lambda $'s and $ n \geq 1$. It is known that

$\displaystyle {\left\Vert {{f^ \ast }} \right\Vert _p} \leq {K_p}{\left\Vert {{g^ \ast }} \right\Vert _p},\quad 1 \leq p < \infty ,$

for some constant $ {K_p}$ depending only on p. We show that $ {K_p} = O(p)$. This will be obtained via a new version of Rosenthal's inequality which generalizes a result of Pinelis and which may be of independent interest.

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Keywords: Moment inequalities, martingale, tangent sequences
Article copyright: © Copyright 1994 American Mathematical Society

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