On the rate of convergence in the central limit theorem for martingales with discrete and continuous time (Q1099875)

Let \((X_ i)_{i\in {\mathbb{N}}}\) be a sequence of real-valued r.v. forming a square integrable martingale difference sequence with respect to the \(\sigma\)-fields \({\mathcal F}_ 0\subset {\mathcal F}_ 1\subset...\subset {\mathcal F}_ i\), i.e. \(X_ i\) is \({\mathcal F}_ i\)-measurable and \(E(X_ i| {\mathcal F}_{i-1})=0\). For \(S_ n=\sum^{n}_{i=1}X_ i\) and the normal d.f. \(\phi\), the author proves the following theorem: For any \(\delta >0\) there exists a finite constant \(C_{\delta}\) such that \[ \sup_{x\in {\mathbb{R}}}| P(S_ n\leq x)-\phi (x)| \leq C_{\delta}(L_{n,2\delta} + N_{n,2\delta})^{1/(3+2\delta)}, \] where \[ L_{n,2\delta}\equiv \sum^{n}_{i=1}E(| X_ i|^{2+2\delta}) \] and \[ N_{n,2\delta}\equiv E(| \sum^{n}_{i=1}E(X^ 2_ i| {\mathcal F}_{i-1})- 1|^{1+\delta}). \] This extends a result of \textit{C. C. Heyde} and \textit{B. M. Brown}, Ann. Math. Statistics 41, 2161-2165 (1970; Zbl 0225.60026), who proved this estimate for \(0<\delta \leq 1\). An example is constructed demonstrating that this bound is asymptotically exact for all \(\delta >0\). The result for discrete-time martingales is then used to derive the corresponding bound on the rate of convergence in the central limit theorem for locally square integrable martingales with continuous time.