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

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 \[ P({d_n} \geq \lambda |{\mathcal {F}_{n - 1}}) = P({e_n} \geq \lambda |{\mathcal {F}_{n - 1}})\] for all real $\lambda$’s and $n \geq 1$. It is known that \[ {\left \| {{f^ \ast }} \right \|_p} \leq {K_p}{\left \| {{g^ \ast }} \right \|_p},\quad 1 \leq p < \infty ,\] 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.