I would like to understand the Almost Sure Invariance Principle:
"We say that the functions f_i: [a,b] →ℝ satisfy the Almost Sure Invariance Principle with error exponent γ < 1/2 if there are a probability space supporting a Brownian motion B and a sequence ξ_i, i ≥ 1, such that
(1) {f_i}_{i≥1}...