Small Deviations of Weighted Fractional Processes and Average Non–linear Approximation Mikhail Lifshits and Werner Linde
We investigate the small deviation problem for weighted fractional Brownian motions in Lq –norm, 1 ≤ q ≤ ∞. Let B H be a fractional Brownian motion with Hurst index 0 < H < 1. If 1/r := H + 1/q, then our main result asserts
lim ε1/H log P
ρ B H
ε→0
1/H
Lq (0,∞)
< ε = −c(H, q) · kρkLr (0,∞)
provided the weight function ρ satisfies a condition slightly stronger than the r–integrability. Thus we extend earlier results for Brownian motion, i.e. H = 1/2, to the fractional case. Our basic tools are entropy estimates for fractional integration operators, a non–linear approximation technique for Gaussian processes as well as sharp entropy estimates for `q –sums of linear operators defined on a Hilbert space.
Small Deviations of Weighted Fractional Processes and ...
provided the weight function Ï satisfies a condition slightly stronger than the râintegrability. Thus we extend earlier results for Brown- ian motion, i.e. H = 1/2, to the fractional case. Our basic tools are entropy estimates for fractional integration operators, a nonâlinear approximation technique for Gaussian processes as well ...
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A i k,l,n. We claim that, for 1 ⤠i ⤠5, we have. ânâ1 k,l=0 |Ai k,l,n| = o(n2â4H ). Let us first consider the case where i = 1. By using Lemma 2.1, we deduce that n.
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