It is shown that ∑ⁿᵢ₌₁ Xₙ and maxⁿᵢ₌₁Xᵢ are asymptotically independent if ᵢ\ is strongly mixing and ∑ⁿᵢ₌₁ Xᵢ is asymptotically Gaussian. This generalizes a result of Anderson and Turkman.
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Tailen Hsing (1995) studied this question.