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Bad rates of convergence for the central limit theorem in Hilbert space. (English) Zbl 0545.60014
The authors show that one can smoothly renorm the Hilbert space $$\ell^ 2$$ such that the rate of convergence in the central limit theorem becomes very poor. Their result reads: Let $$\epsilon>0$$ and a real sequence $$\xi_ n\to 0$$, $$n\to \infty$$, be given. There is a norm $$N(\cdot)$$ on $$\ell^ 2$$ and a sequence $$(X_ i)_{i\in {\mathbb{N}}}$$ of i.i.d. $$\ell^ 2$$-valued r.v. such that the following holds:
(a) $$(1-\epsilon)\| x\| \leq N(x)\leq \| x\|$$; $$x\in \ell^ 2;$$
(b) $$N(\cdot)$$ is infinitely many times differentiable, and each of its differentials is bounded on the unit sphere;
(c) For infinitely many values of n, $$\sup_{t\geq 0}| P\{N(n^{- {1\over2}}\sum^{n}_{i=1}X_ i)\leq t\}-\gamma \{x;N(x)<t\}| \geq \xi_ n$$, where $$\gamma$$ is the Gaussian measure with the same covariance as $$X_ i.$$
This result is based on the fact that for $$N(\cdot)$$ fails the following condition that the usual norm $$\| \cdot \|$$ fulfills, namely, $$\gamma \{s\leq \| x\| \leq t\}\leq C(t-s)$$ for a constant C and all $$0\leq s<t$$.
Reviewer: L.Hahn

##### MSC:
 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) 46B20 Geometry and structure of normed linear spaces
##### Keywords:
bad rates; Hilbert space; rate of convergence; Gaussian measure
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