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

Let (Xₙ) ₍ ₗ ₉ be a sequence of r. v. 's with E Xₙ = 0, E (ⁿ₈ = ₁ Xᵢ) ²/n ² > 0, ₍, ₌E (^m + n₈ = ₌ + ₁ Xᵢ) ²/n 1, or (k) = O (b^-k) for some b > 1, we obtain functions f_ (n) such that \|Xₙ\|_ = o (f_ (n) ) for some (2, is sufficient for the functional c. l. t. , but the c. l. t. may fail to hold if \|Xₙ\|_ = O (f_ (n) ).