An Erdös–Révész type law of the iterated logarithm for reflected fractional Brownian motion

Abstract

Let B H = {B H (t) : t ∈ R} be a fractional Brownian motion with Hurst parameter H ∈ (0, 1). For the stationary storage processwe provide a tractable criterion for assessing whether, for any positive, non-decreasing function f , P(Q B H (t) > f (t) i.o.) equals 0 or 1. Using this criterion we find that, for a family of functions

Sample path properties of reflected Gaussian processes

“…and that the length of the interval h p (t) is smallest possible. Theorem 1.1 and Theorem 1.3 generalize the main results of Dębicki and Kosiński (2017), which considered the special case when X ≡ B H is a fractional Brownian motion with any Hurst parameter H ∈ (0, 1); see also (Shao, 1992;Dębicki and Kosiński, 2018) for similar results for non-reflected Gaussian processes and Gaussian order statistics. The organization of the rest of paper is as follows.…”