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Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

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  • Pilipauskaitė, Vytautė
  • Surgailis, Donatas

Abstract

We discuss joint temporal and contemporaneous aggregation of N independent copies of AR(1) process with random-coefficient a∈[0,1) when N and time scale n increase at different rate. Assuming that a has a density, regularly varying at a=1 with exponent −1<β<1, different joint limits of normalized aggregated partial sums are shown to exist when N1/(1+β)/n tends to (i) ∞, (ii) 0, (iii) 0<μ<∞. The limit process arising under (iii) admits a Poisson integral representation on (0,∞)×C(R) and enjoys ‘intermediate’ properties between fractional Brownian motion limit in (i) and sub-Gaussian limit in (ii).

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  • Pilipauskaitė, Vytautė & Surgailis, Donatas, 2014. "Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1011-1035.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:2:p:1011-1035
    DOI: 10.1016/j.spa.2013.10.004
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    References listed on IDEAS

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    1. Leipus, Remigijus & Paulauskas, Vygantas & Surgailis, Donatas, 2005. "Renewal regime switching and stable limit laws," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 299-327.
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    4. Granger, C. W. J., 1980. "Long memory relationships and the aggregation of dynamic models," Journal of Econometrics, Elsevier, vol. 14(2), pages 227-238, October.
    5. Gaigalas, Raimundas, 2006. "A Poisson bridge between fractional Brownian motion and stable Lévy motion," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 447-462, March.
    6. Zaffaroni, Paolo, 2007. "Aggregation and memory of models of changing volatility," Journal of Econometrics, Elsevier, vol. 136(1), pages 237-249, January.
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    Cited by:

    1. Puplinskaitė, Donata & Surgailis, Donatas, 2015. "Scaling transition for long-range dependent Gaussian random fields," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2256-2271.
    2. Pilipauskaitė, Vytautė & Surgailis, Donatas, 2015. "Joint aggregation of random-coefficient AR(1) processes with common innovations," Statistics & Probability Letters, Elsevier, vol. 101(C), pages 73-82.
    3. Doukhan, Paul & Jakubowski, Adam & Lopes, Silvia R.C. & Surgailis, Donatas, 2019. "Discrete-time trawl processes," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1326-1348.
    4. Kern, Peter & Wedrich, Lina, 2015. "Dilatively semistable stochastic processes," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 101-108.
    5. Surgailis, Donatas, 2020. "Scaling transition and edge effects for negatively dependent linear random fields on Z2," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7518-7546.
    6. Nedényi, Fanni & Pap, Gyula, 2016. "Iterated scaling limits for aggregation of random coefficient AR(1) and INAR(1) processes," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 16-23.
    7. Pilipauskaitė, Vytautė & Surgailis, Donatas, 2017. "Scaling transition for nonlinear random fields with long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2751-2779.
    8. Bhatti, T. & Kern, P., 2017. "An integral representation of dilatively stable processes with independent increments," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 209-227.
    9. Surgailis, Donatas, 2024. "Scaling limits of nonlinear functions of random grain model, with application to Burgers’ equation," Stochastic Processes and their Applications, Elsevier, vol. 174(C).

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