Another view of the CLT in Banach spaces (Q960187)

The paper under review gives an extensive and elegant study of the central limit theorem (CLT) in Banach spaces when original data are allowed slight modifications. Let \(X\) be a random variable taking values in a separable Banach space \(B\) with distribution \(p\) and such that \(Ef(X)= 0\), \(Ef^2(X)<\infty\) for every linear functional \(f\) on \(B\). Then there is a Hilbert space \(H_\mu\) determined by the covariance of \(\mu\) such that \(H_\mu\subset B\). Furthermore, for all \(\varepsilon> 0\) and \(x\) in the \(B\)-norm closure \(\overline H_\mu\) of \(H_\mu\), there is a unique point, \(T_\varepsilon(x)\in H_\mu\), with minimal \(H_\mu\)-norm in the \(B\)-norm ball of radius \(\varepsilon\) and center \(x\). In a variety of settings there are obtained CLTs for \(T_\varepsilon(X)\) and its certain further modifications, even when \(X\) itself fails CLT. The motivation for the use of \(T_\varepsilon\) comes from the large deviation rates for Gaussian measures. One of the results provides a general and universal improvement in the CLT. It states that under the above assumptions if \(\overline H_\mu= B\) and \(\mu\) has weak-star sequentially continuous covariance function then \(T_{\psi(\varepsilon)(\| T_\varepsilon(X)\|_\mu\vee 1)}(T_\varepsilon(X))\) satisfies CLT for every \(\varepsilon> 0\) and every increasing continuous function \(\psi\) with \(\psi(0)= 0\) and \(\psi(t)\leq t\) on \([0,\infty)\). Here \(\|\cdot\|_\mu\) denotes \(H_\mu\)-norm. In particular, when \(E\| X\|^2<\infty\) then the above theorem applies. Several examples involving \(B\)-valued random series show, among others, that the assumptions in the above theorem are close to being necessary.