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Operator self-similar processes and functional central limit theorems

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  • Characiejus, Vaidotas
  • Račkauskas, Alfredas

Abstract

Let {Xk:k≥1} be a linear process with values in the separable Hilbert space L2(μ) given by Xk=∑j=0∞(j+1)−Dεk−j for each k≥1, where D is defined by Df={d(s)f(s):s∈S} for each f∈L2(μ) with d:S→R and {εk:k∈Z} are independent and identically distributed L2(μ)-valued random elements with Eε0=0 and E‖ε0‖2<∞. We establish sufficient conditions for the functional central limit theorem for {Xk:k≥1} when the series of operator norms ∑j=0∞‖(j+1)−D‖ diverges and show that the limit process generates an operator self-similar process.

Suggested Citation

  • Characiejus, Vaidotas & Račkauskas, Alfredas, 2014. "Operator self-similar processes and functional central limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2605-2627.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:8:p:2605-2627
    DOI: 10.1016/j.spa.2014.03.007
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    References listed on IDEAS

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    1. Laha, R. G. & Rohatgi, V. K., 1981. "Operator self similar stochastic processes in," Stochastic Processes and their Applications, Elsevier, vol. 12(1), pages 73-84, October.
    2. Maejima, Makoto & Mason, J. David, 1994. "Operator-self-similar stable processes," Stochastic Processes and their Applications, Elsevier, vol. 54(1), pages 139-163, November.
    3. Cremers, Heinz & Kadelka, Dieter, 1986. "On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in LEP," Stochastic Processes and their Applications, Elsevier, vol. 21(2), pages 305-317, February.
    4. Lavancier, Frédéric & Philippe, Anne & Surgailis, Donatas, 2009. "Covariance function of vector self-similar processes," Statistics & Probability Letters, Elsevier, vol. 79(23), pages 2415-2421, December.
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    Cited by:

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    2. 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.

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