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.

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