  We develop a new method for showing that a given sequence of random variables
verifies an appropriate law of the iterated logarithm. Our tools involve the
use of general estimates on multidimensional Wasserstein distances, that are in
turn based on recently developed inequalities involving Stein matrices and
transport distances. Our main application consists in the proof of the exact
law of the iterated logarithm for the Hermite variations of a fractional
Brownian motion in the critical case.
