The conditional central limit theorem in Hilbert spaces

In this paper, we give necessary and sufficient conditions for a stationary sequence of random variables with values in a separable Hilbert space to satisfy the conditional central limit theorem introduced in Dedecker and Merlevède (Ann. Probab. 30 (2002) 1044–1081). As a consequence, this theorem implies stable convergence of the normalized partial sums to a mixture of normal distributions. We also establish the functional version of this theorem. Next, we show that these conditions are satisfied for a large class of weakly dependent sequences, including strongly mixing sequences as well as mixingales. Finally, we present an application to linear processes generated by some stationary sequences of H-valued random variables.