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1 results

  • On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise

  • Part of
  • Remigijus Leipus, Vygantas Paulauskas, Donatas Surgailis
  • Journal:
    Journal of Applied Probability / Volume 43 / Issue 2 / June 2006
    Published online by Cambridge University Press:
    14 July 2016, pp. 421-440
    Print publication:
    June 2006
    • Article
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  • We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation Xt = atXt−1 + εt with random (renewal-reward) coefficient, at, taking independent, identically distributed values Aj ∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, εt, belonging to the domain of attraction of an α-stable law (0 < α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of Aj near the unit root a = 1, we show that the partial sums process of Xt converges to a λ-stable Lévy process with index λ < α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance Xt to that of infinite-variance Xt.