On the Joint Limiting Distribution of Sums and Maxima of Stationary Normal Sequence

Let X₁, X₂, … be a stationary sequence of standard normal random variables. Let ρₙ = E(X₁ Xₙ₊₁). Ho and Hsing derived the asymptotic joint distribution of ∑i = 1ⁿ Xᵢ and max1 ≤ i ≤ n Xᵢ for the case ρₙ log n γ ∈ [0, ∞). In this paper we extend this result for the case where ρₙ is convex with ρₙ = o(1), and (ρₙ log n)⁻¹ is monotone with (ρₙ log n)⁻¹ = o(1).