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Point processes associated with stationary stable processes

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  • Resnick, Sidney
  • Samorodnitsky, Gennady

Abstract

Point processes induced by stationary symmetric [alpha]-stable (S[alpha]S) processes can have diverse behavior. We distinguish two cases, depending on whether the stationary S[alpha]S process is governed by a dissipative or conservative flow. In the case of dissipative flows, the process is a mixed moving average and the family of scaled point processes converge to a Poisson cluster process. In the case of a conservative flow, we give two examples showing how diverse the behavior can be.

Suggested Citation

  • Resnick, Sidney & Samorodnitsky, Gennady, 2004. "Point processes associated with stationary stable processes," Stochastic Processes and their Applications, Elsevier, vol. 114(2), pages 191-209, December.
    • Handle: RePEc:eee:spapps:v:114:y:2004:i:2:p:191-209
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    References listed on IDEAS

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    1. Resnick, Sidney & Samorodnitsky, Gennady & Xue, Fang, 2000. "Growth rates of sample covariances of stationary symmetric [alpha]-stable processes associated with null recurrent Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 321-339, February.
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    Cited by:

    1. Charlot, F. & Rachdi, M., 2008. "On the statistical properties of a stationary process sampled by a stationary point process," Statistics & Probability Letters, Elsevier, vol. 78(4), pages 456-462, March.
    2. Panigrahi, Snigdha & Roy, Parthanil & Xiao, Yimin, 2021. "Maximal moments and uniform modulus of continuity for stable random fields," Stochastic Processes and their Applications, Elsevier, vol. 136(C), pages 92-124.
    3. Raluca M. Balan & Sana Louhichi, 2009. "Convergence of Point Processes with Weakly Dependent Points," Journal of Theoretical Probability, Springer, vol. 22(4), pages 955-982, December.
    4. Fasen, Vicky & Roy, Parthanil, 2016. "Stable random fields, point processes and large deviations," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 832-856.
    5. Parthanil Roy, 2017. "Maxima of stable random fields, nonsingular actions and finitely generated abelian groups: A survey," Indian Journal of Pure and Applied Mathematics, Springer, vol. 48(4), pages 513-540, December.

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