Small deviations for fractional stable processes

Mikhail Lifshits, Thomas Simon

Research output

28 Citations (Scopus)

Abstract

Let {Rt, 0 ≤ t ≤ 1} be a symmetric α-stable Riemann-Liouville process with Hurst parameter H > 0. Consider a translation invariant, β-self-similar, and p-pseudo-additive functional semi-norm ∥·∥. We show that if H > β + 1/p and γ = (H - β - 1/p)-1, then lim ε↓0 εγ log ℙ[∥ R ∥ ≤ ε] = -K ∈ [-∞, 0), with K finite in the Gaussian case α = 2. If α < 2, we prove that K is finite when R is continuous and H > β + 1/p + 1/α. We also show that under the above assumptions, lim ε↓0 εγ log ℙ[∥ X ∥ ≤ ε] = -K ∈ (-∞, 0), where X is the linear & alpha;-stable fractional motion with Hurst parameter H ∈ (0, 1) (if α = 2, then X is the classical fractional Brownian motion). These general results cover many cases previously studied in the literature, and also prove the existence of new small deviation constants, both in Gaussian and non-Gaussian frameworks.

Original languageEnglish
Pages (from-to)725-752
Number of pages28
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume41
Issue number4
DOIs
Publication statusPublished - 1 Jul 2005

Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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