Comparison theorems for the small ball probabilities of the Green Gaussian processes in weighted $L_2$-norms

Abstract

Comparison theorems are proved for small ball probabilities of the Green Gaussian processes in weighted L 2 -norms. The sharp small ball asymptotics are found for many classical processes under quite general assumptions on the weight.2010 Mathematics Subject Classification. Primary 60G15.

‐small Deviations for Weighted Stationary Processes

“…Remark 2.3 In fact, the sharp small ball asymptotics for these processes were obtained in [19] and [24], see also [21] for more general weights. However, this is strongly connected with the fact that β (ω) and B {m} are the Green Gaussian processes i.e.…”

“…Many studies have been devoted to the asymptotic behavior of its L 2 -small deviation probabilities P ||Y || 2 2 = T |Y (t)| 2 m(dt) ≤ ε 2 , as ε → 0, see e.g. [7,8,9,18,19,20,21,24], to mention just a small sample. The importance of small deviation probabilities in a broader context and a large number of their applications are described in the surveys [14,15]; for an extensive up-to-date bibliography see [16].…”

“…as ε → 0, see e.g. [7,8,9,18,19,20,21,24], to mention just a small sample. The importance of small deviation probabilities in a broader context and a large number of their applications are described in the surveys [14,15]; for an extensive up-to-date bibliography see [16].…”