Self-stabilizing processes

K. J. Falconer, J. Lévy Vehel

Research output: Contribution to journalArticlepeer-review

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We construct "self-stabilizing" processes {Z(t), t ∈[t0,t1)}. These are random processes which when "localized", that is scaled around t to a fine limit, have the distribution of an α(Z(t))-stable process, where α is some given function on ℝ. Thus the stability index at t depends on the value of the process at t. Here we address the case where α: ℝ → (0,1). We first construct deterministic functions which satisfy a kind of autoregressive property involving sums over a plane point set Π. Taking Π to be a Poisson point process then defines a random pure jump process, which we show has the desired localized distributions.
Original languageEnglish
Pages (from-to)409-434
Number of pages26
JournalStochastic Models
Issue number4
Early online date11 Nov 2018
Publication statusPublished - 2018


  • Local form
  • Self-stabilizing
  • Stable process


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