A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables

Let (Xₙ)n ∈ X J be a sequence of r.v.'s with E Xₙ = 0, E(∑ⁿi = 1 Xᵢ)²/n → σ² > 0, n,mE(∑m + ni = m + 1 Xᵢ)²/n < ∞. We prove the functional c.l.t. for (Xₙ) under assumptions on αₙ(k) = \|P(A ∩ B) - P(A)P(B)|:A ∈ σ(Xᵢ: 1 ≤ i ≤ m), B ∈ σ(Xᵢ: m + k ≤ i ≤ n), 1 ≤ m ≤ n - k\ and the asymptotic behaviour of \|Xₙ\|_β for some β ∈ (2, ∞. For the special cases of strongly mixing sequences (Xₙ) with α(k) = αₙ(k) = O(k⁻ᵃ) for some $a > 1$, or α(k) = O(b⁻ᵏ) for some $b > 1$, we obtain functions f_β(n) such that \|Xₙ\|_β = o(f_β(n)) for some β ∈ (2, ∞ is sufficient for the functional c.l.t., but the c.l.t. may fail to hold if \|Xₙ\|_β = O(f_β(n)).