The conditional central limit theorem in Hilbert spaces

Jérôme Dedecker and Florence Merlevède

Stochastic Processes and their Applications, 2003, vol. 108, issue 2, 229-262

Abstract: 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 -valued random variables.

Keywords: Hilbert; space; Central; limit; theorem; Weak; invariance; principle; Strictly; stationary; process; Stable; convergence; Strong; mixing; Mixingale; Linear; processes (search for similar items in EconPapers)
Date: 2003
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Citations: View citations in EconPapers (8)

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