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Non-causal strictly stationary solutions of random recurrence equations

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  • Brandes, Dirk-Philip
  • Lindner, Alexander

Abstract

Let (Mn,Qn)n∈N be an i.i.d. sequence in R2. Much attention has been paid to causal strictly stationary solutions of the random recurrence equation Xn=MnXn−1+Qn, n∈N, i.e. to strictly stationary solutions of this equation when X0 is assumed to be independent of (Mn,Qn)n∈N. Goldie and Maller (2000) gave a complete characterisation when such causal solutions exist. In the present paper we shall dispose of the independence assumption of X0 and (Mn,Qn)n∈N and derive necessary and sufficient conditions for a strictly stationary, not necessarily causal solution of this equation to exist.

Suggested Citation

  • Brandes, Dirk-Philip & Lindner, Alexander, 2014. "Non-causal strictly stationary solutions of random recurrence equations," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 113-118.
    • Handle: RePEc:eee:stapro:v:94:y:2014:i:c:p:113-118
    DOI: 10.1016/j.spl.2014.06.027
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    References listed on IDEAS

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    1. Lindner, Alexander & Maller, Ross, 2005. "Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1701-1722, October.
    2. Behme, Anita & Lindner, Alexander & Maller, Ross, 2011. "Stationary solutions of the stochastic differential equation with Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 91-108, January.
    3. Lanne Markku & Saikkonen Pentti, 2011. "Noncausal Autoregressions for Economic Time Series," Journal of Time Series Econometrics, De Gruyter, vol. 3(3), pages 1-32, October.
    4. Peter J. Brockwell & Alexander Lindner, 2010. "Strictly stationary solutions of autoregressive moving average equations," Biometrika, Biometrika Trust, vol. 97(3), pages 765-772.
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