Limiting distribution of the continuity modulus for Gaussian processes with stationary increments

Let {X(t),t∈R}{X(t),t∈R} be a Gaussian process with stationary increments, zero mean and a.s. continuous paths, whose variogram γ(t)γ(t) behaves like c|t|αc|t|α, c>0c>0, α∈(0,2)α∈(0,2), as t→0t→0. We show that the continuity modulus of XX has asymptotically Gumbel distribution. In the case α=2α=2, a non-Gumbel limiting distribution is obtained.