## On the rate of convergence in the central limit theorem for martingales with discrete and continuous time.(English)Zbl 0639.60030

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 C. C. Heyde and 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.
Reviewer: L.Hahn

### MSC:

 60F05 Central limit and other weak theorems 60G42 Martingales with discrete parameter 60G44 Martingales with continuous parameter

Zbl 0225.60026
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