Small Deviation Probabilities of Sums of Independent Random Variables

Comparison Theorems for Small Deviations of Random Series

“…By using Theorem 1, Dunker, Lifshits and Linde [5] obtain similar results when the random variables satisfy the following additional condition:…”

“…The advantage of the results in [5] over Theorem 1 is that the asymptotic behavior of the small deviation probability of V is expressed (implicitly) in terms of the Laplace transform of ξ 1 instead of the Laplace transform of V .…”

“…Typically, b n = φ(n) so that one can compute the small deviation asymptotics on the righthand side of (2) by using the results of [5]. {b n } can also be chosen so that one can find the exact expression of the Laplace transform of the corresponding series using the method of [8].…”

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.…”

Точная Асимптотика Малых Уклонений Для Ряда Броуновских Функционалов

“…[43], [30, с. 449]) μ(m) law = 2μ(|B|), где правая часть совпадает с правой частью (35). Тогда, применив (33), получаем следующее соот-ношение.…”