## On the rate of convergence in the martingale central limit theorem.(English)Zbl 1277.60051

Let $$\underline{X}_n=(X_1,\ldots,X_n)$$ be a square integrable martingale difference sequence of random variables, i.e. $$E(X_i|X_1,\ldots,X_{i-1})=0$$ and $$E(X_i^2)<\infty$$ for all $$i=1,\ldots,n$$, so that $$s^2(\underline{X}_n)=\sum_{i=1}^nE(X_i^2)$$ and $$V^2(\underline{X}_n)=s^{-2}(\underline{X}_n)\sum_{i=1}^n E(X_i^2|X_1,\ldots,X_{i-1})$$ are well-defined. The martingale CLT says that if $$V^2(\underline{X}_n)\rightarrow1$$ in probability as $$n\to\infty$$ and a Lindeberg condition is satisfied, then $$s^{-1}(\underline{X}_n)\sum_{i=1}^nX_i$$ converges in distribution to a standard normal random variable. Typically, bounds on the rate of convergence in this CLT involve the $$L_p$$-norm $$\|V^2(\underline{X}_n)-1\|_p$$ for some $$p\in[1,\infty]$$ in the form $$\|V^2(\underline{X}_n)-1\|_p^{p/(2p+1)}$$. In the present paper it is shown that the exponent $$p/(2p+1)$$ is optimal. This closes a notable gap in the theory of bounds on the rate of convergence in the martingale CLT.

### MSC:

 60F05 Central limit and other weak theorems 60G42 Martingales with discrete parameter
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### References:

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