TY - JOUR

T1 - Small deviations for fractional stable processes

AU - Lifshits, Mikhail

AU - Simon, Thomas

PY - 2005/7/1

Y1 - 2005/7/1

N2 - 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.

AB - 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.

KW - Fractional Brownian motion

KW - Gaussian process

KW - Linear fractional stable motion

KW - Riemann-Liouville process

KW - Small ball constants

KW - Small ball probabilities

KW - Small deviations

KW - Stable process

KW - Wavelets

UR - http://www.scopus.com/inward/record.url?scp=19344365741&partnerID=8YFLogxK

U2 - 10.1016/j.anihpb.2004.05.004

DO - 10.1016/j.anihpb.2004.05.004

M3 - Article

AN - SCOPUS:19344365741

VL - 41

SP - 725

EP - 752

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

SN - 0246-0203

IS - 4

ER -