# Almost sure invariance principle

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1. Apr 13, 2015

### Sol_

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} and {ξ_i}_{i≥1} have the same distribution;
(2) |B(n) - ∑_{i=1}^{n} (ξ_i)| < O(n^γ)
almost surely as n → ∞."

If anyone can give me an interpretation, I'll be very grateful

2. Apr 18, 2015