On the conditional covariance condition in the martingale CLT
Let \(H\) be a separable Hilbert space with scalar product \((\cdot, \cdot)\) and norm \(|\cdot |\). For every integer \(n\), let \(X_{n1}, \dots, X_{nk_n}\) be a square integrable martingale difference sequence of \(H\)-valued random elements with respect to \(\sigma\)-fields \({\mathcal F}_{n0} \subset {\mathcal F}_{n1} \subset \cdots \subset {\mathcal F}_{nk_n}\). The conditional covariance operator \(Q_{ni} : H \to H\) of \(X_{ni}\) given \({\mathcal F}_{n,i - 1}\) is defined by \(Q_{ni} x = E[(X_{ni}, x)X_{ni} \mid {\mathcal F}_{n, i - 1}]\) for \(x \in H\) and \(i = 1, \dots, k_n\). Let \(Y\) be a mean zero Gaussian random element in \(H\) with covariance operator \(Q\). One form of the central limit theorem for martingales in Hilbert spaces states that \(S_{nk_n} = \sum^{k_n}_{i = 1} X_{ni}\) converges in distribution to \(Y\) provided that a conditional version of the classical Lindeberg condition is satisfied and, in addition, that the conditional covariance condition \[ \sum^{k_n}_{i = 1} Q_{ni} \to Q \quad \text{in probability as} \quad n \to \infty \tag{C} \] holds. Clearly, the accuracy of the approximation of the law of \(S_{nk_n}\) by that of \(Y\) will depend on the distance between \(\sum^{k_n}_{k = 1} Q_{ni}\) and \(Q\). The present paper develops a method to incorporate this distance into bounds on the distance of the laws of \(S_{nk_n}\) and \(Y\) provided that such bounds are available already under rather strong requirements on the \(Q_{ni}\) like \(\sum^{k_n}_{i = 1} Q_{ni} = Q\) almost surely. The method is based on the newly introduced concept of stopping projections which can be used to control the behavior of the \(Q_{ni}\) and is appropriate for dealing with discrete time martingale differences taking their values in finite-dimensional as well as infinite-dimensional Hilbert spaces.
scientific article; zbMATH DE number 846599