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Limit theorems for Hilbert space-valued linear processes under long range dependence

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  • Düker, Marie-Christine

Abstract

Let (Xk)k∈Z be a linear process with values in a separable Hilbert space H given by Xk=∑j=0∞(j+1)−Nεk−j for each k∈Z, where N:H→H is a bounded, linear normal operator and (εk)k∈Z is a sequence of independent, identically distributed H-valued random variables with Eε0=0 and E‖ε0‖2<∞. We investigate the central and the functional central limit theorem for (Xk)k∈Z when the series of operator norms ∑j=0∞‖(j+1)−N‖op diverges. Furthermore, we show that the limit process in case of the functional central limit theorem generates an operator self-similar process.

Suggested Citation

  • Düker, Marie-Christine, 2018. "Limit theorems for Hilbert space-valued linear processes under long range dependence," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1439-1465.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:5:p:1439-1465
    DOI: 10.1016/j.spa.2017.07.015
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