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The conditional central limit theorem in Hilbert spaces. (English) Zbl 1075.60501

The conditional central limit theorem of the authors [Ann.Probab.30, 1044–1081 (2002; Zbl 1015.60016)] is studied for dependent random variables. For a measure preserving transformation \(T\) on a Hilbert space \(\mathbf H\) the sequence \(X_i = X_0\circ T^i\) and the process \(S_n=X_1 + \dots + X_n\) are defined. In the first part of the paper equivalent conditions for the conditional CLT for \(S_n\) and for the functional CLT for \(W_n(t) = S_{[nt]} + (nt-[nt])X_{[nt]+1}\) are found. In the second part the results are applied to weakly dependent sequences and to \(\mathbf H\)-valued linear processes.

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles

Citations:

Zbl 1015.60016
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