Bad Rates of Convergence for the Central Limit Theorem in Hilbert Space

Abstract

We show that one can smoothly renorm the Hilbert space $H$ such that the rate of convergence in the central limit theorem becomes very bad. More precisely, let us fix a sequence $\xi_n \rightarrow 0$ and $\varepsilon > 0$. We can then construct a norm $N(\cdot)$ on the Hilbert space, and a bounded random variable $X$ on $H$ with the following properties: (a) The norm $N(\cdot)$ is $(1 + \varepsilon)$ equivalent to the usual norm. It is infinitely many times differentiable, and each differential is bounded on the unit sphere. (b) If $(X_i)$ denotes independent copies of $X$, and if $\gamma$ is the Gaussian measure with the same covariance as $X$, then the inequality $\operatorname{Sup}_{t>0}|P\{N(n^{-1/2} \sum^n_{i=1} X_i) \leq t\} - \gamma\{x; N(x) \leq t\}| \geq \xi_n$ occurs for infinitely many $n$.