On the iterated martingale transforms (Q2725503)

Let \(f= (f_n, n\geq 0)\) be an \((P,\mathbb{F})\)-martingale on some filtered probability space \((\Omega,{\mathcal F},\mathbb{F}, P)\), \(I^{(m)}_n= \sum^n_{j=1} I^{(m-1)}_{j-1}(f_j- f_{j-1})\), \(I^{(m)}_0= 0\), \(n\geq 0\), \(m\geq 0\), where \(I^{(0)}_n\equiv 1\) and \(I^{(1)}_n= f_n\), \(n\geq 0\). It is proved that for \(1\leq p<\infty\) and \(m\geq 1\), NEWLINE\[NEWLINE\Biggl\|\sum_{n\geq 0}|I^{(m)}_n|\Biggr\|_p\leq (4mp)^m\Biggl\|\sum^\infty_{j=1} (f_j- f_{j-1})^2\Biggr\|_p.NEWLINE\]NEWLINE For a continuous martingale \(M\) approximation in probability of \(H_m(M,\langle M\rangle)\), \(m\geq 1\), by the iterated martingale transforms is also considered, where \(H_m(x,y)\) is the Hermite polynomial with parametric variable \(y> 0\).