Logarithmic L2-Small Ball Asymptotics for some Fractional Gaussian Processes

Small ball probabilities for Gaussian random fields and tensor products of compact operators

“…Latest results on L 2 -small deviation asymptotics for onedimensional Gaussian random functions related to Brownian motion can be found in [BNO], [GHT]- [GHLT2], [Na1]- [NaNi1]. Some advancements connected to fractional Brownian motion can be found in [Br] and [NaNi2].…”

“…In [NaNi2] the logarithmic small deviation asymptotics was described for fractional and ordinary Lévy's random fields on arbitrary domain Ω ⊂ R d . According to the well-known Karhunen -Loève expansion which is valid also in the multiparameter case, see [Ad], we have in distribution…”

Log-level comparison principle for small ball probabilities

“…For the power-type decreasing of λ n the one-term asymptotics of N (λ) as λ → 0 provides the one-term asymptotics of λ n . So, Proposition 2.1 [NN2], Theorem 4.2 [KNN] and Theorem 4.2 [Na2] are particular cases of this statement. But for the super-power decreasing of λ n the condition (1.5) is weaker than λ n ∼ λ n .…”

“…So, we need so-called logarithmic asymptotics, that is the asymptotics of ln P{||X|| µ ≤ ε} as ε → 0. It was shown in [NN2] (see also [KNN] and [Na2]) that in some cases this asymptotics is completely determined by the one-term asymptotics of λ n . This gives a logarithmic version of the comparison theorem.…”

Entropy-constrained functional quantization of Gaussian processes

“…The eigenvalue problem has been solved for several other Gaussian processes and other Hilbert spaces including stationary processes with regular highfrequency behaviour of the spectral density, fractional Brownian sheets, Lévy's fractional Brownian motion, integrated Gaussian processes and Gaussian martingales. See [15], [14, p. 79], [11], [2], [9], [3], [12], [13].…”