Central limit theorem under the Dedecker-Rio condition in some Banach spaces (Q6615467)

Let \((X_i)_{i\in\mathbb{Z}}\) be an ergodic and stationary sequence of centred random variables taking values in a 2-smooth Banach space with a Schauder basis, adapted to a non-decreasing and stationary filtration \((\mathcal{F}_i)_{i\in\mathbb{Z}}\). Writing \(S_n=X_1+\cdots+X_n\), and assuming suitable convergence of \(\lVert X_0\rVert\mathbb{E}[S_n|\mathcal{F}_0]\), the author establishes convergence of the distribution of \(\frac{1}{\sqrt{n}}S_n\) to Gaussian as \(n\to\infty\), thus extending the Dedecker-Rio central limit theorem. This setting includes \(L^p(\mu)\)-valued random variables for \(2\leq p<\infty\) and a \(\sigma\)-finite real measure \(\mu\). Applications to empirical processes are discussed, and various sufficient conditions for this central limit theorem are given.