Exact Small Ball Constants for Some Gaussian Processes under the L 2-Norm

Abstract

We find some logarithmic and exact small deviation asymptotics for the L 2 -norms of certain Gaussian processes closely connected with a Wiener process. In particular, processes obtained by centering and integrating Brownian motion and Brownian bridge are examined. Bibliography: 28 titles.

Exact L 2 -small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems

“…This technique was already used in [17] for various one time integrated processes. Temporarily let assume that in (6.1) ϑ = 1, later we will incorporate the influence of ϑ in the final formula.…”

“…In [17] the final result for the one-fold integrated Wiener process was obtained among a series of similar asymptotics. Namely, it was proved that P {||W 1 || 2 ≤ ε} ∼ 1.414... · (8/ √ 3π) ε 1/3 exp(−(3/8)ε −2/3 ).…”

Precise small deviations in L 2 of some Gaussian processes appearing in the regression context

“…The methods to obtain the precise asymptotics of (1) for more complicated Gaussian processes have been gradually polished in the papers by Li [17], Dunker, Lifshits and Linde [5], Fill and Torcaso [9], Gao, Hannig, Lee and Torcaso [10][11][12][13], Beghin, Nikitin and Orsingher [3], Nazarov and Nikitin [26], Nazarov [23][24][25], Kharinski and Nikitin [16], and Nikitin and Pusev [27]. However, this problem is completely solved only in rare cases, and advances for new examples of Gaussian processes are certainly of interest.…”

“…Lifshits himself has never published it, so we cite its formulation without proof from [3] and [26]. See also the recent paper [6] for a detailed proof.…”

Об Одном Семействе Преобразований Гауссовских Случайных Функций

“…Теорема 3 доказана. В примерах, рассмотренных выше, асимптотика малых уклонений для исходных процессов хорошо известна (процессы W и B являются классическими, процесс W 1 разобран в [3]). Применяя теоремы 1 и 3, получаем следующее предложение.…”