Small Deviations in a Space of Trajectories

The small-time Chung-Wichura law for Lévy processes with non-vanishing Brownian component

“…Let α ∈ (0, 2] and U (α) = (U (α) (t)) t≥0 be a symmetric stable Lévy process of index α. As shown by Mogul'skii [25], there exists a constant u α ∈ (0, ∞) such that…”

“…25) with C (n −n ) as in (5.9) and N m = {g ∈ M : a k < g(s k ) ≤ b k , 1 ≤ k ≤ m}. (5.26) Now the proof of the lemma is completed as follows: let f ∈ (σ π/ √ 8)K and ε > 0.…”

Small deviations of iterated processes in the space of trajectories

“…Suppose that the distributions of (η 1 + η 2 + · · · + η )/B converge weakly to a strictly stable distribution G α , α ∈ (0 2], with G α ((−∞ 0)) ∈ (0 1), where {B } is a sequence of positive constants. Applying [14,Theorem 4], we arrive at (10) for homogeneous processes with independent increments and, consequently, for strictly stable processes ξ( ) such that ξ(1) has distribution G α . For α = 2, ξ( ) is the Brownian motion.…”

“…One of the most important studied classes are sums of independent random variables and stochastic processes with independent increments. Mogul'skii [14] has studied the asymptotic behavior of log Q as → ∞. Further results in this direction may be found in Borovkov and Mogul'skii [3] and references therein.…”

Donsker’s Invariance Principle Under the Sub-linear Expectation with an Application to Chung’s Law of the Iterated Logarithm

“…for all n sufficient large. A more general inequality of this type is the following small deviation obtained by Mogul'skiǐ (1974): if 0 < x n → 0 and n 1/2 x n → ∞, then log P max i≤n |S i | ≤ n 1/2 x n ∼ − π 2 8x 2 n .…”