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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
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