8 research outputs found
Small deviations for fractional stable processes
Let R be a symmetric a-stable Riemann-Liouville process with Hurst parameter
H > 0. Consider ||.|| a translation invariant, b-self-similar, and
p-pseudo-additive functional semi-norm. We show that if H > (b + 1/p) and c =
(H - b - 1/p), then x log P [ log ||R|| - k 0,
with k finite in the Gaussian case a = 2. If a < 2, we prove that k is finite
when R is continuous and H > (b + 1/p + 1/a). We also show that under the above
assumptions, x log P [ log ||X|| - k 0, where k is
finite and X is the linear a-stable fractional motion with Hurst parameter 0 <
H < 1 (if a = 2, then X is the classical fractional Brownian motion). These
general results recover many cases previously studied in the literature, and
also prove the existence of new small deviation constants, both in Gaussian and
Non-Gaussian frameworks.Comment: 30 page
Small deviations of weighted fractional processes and average non-linear approximation
We investigate the small deviation problem for weighted fractional Brownian motions in Lq–norm, 1 ≤ q ≤∞.LetBHbea fractional Brownian motion with Hurst index 0 <H<1. If 1/r: = H +1/q, then our main result asserts lim ε→0 ε1/H ����ρB H log