On Jiang's asymptotic distribution of the largest entry of a sample correlation matrix

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Let {X, Xₖ,ᵢ; i ≥ 1, k ≥ 1 } be a double array of nondegenerate i.i.d. random variables and let {pₙ; n ≥ 1 } be a sequence of positive integers such that n/pₙ is bounded away from 0 and ∞. This paper is devoted to the solution to an open problem posed in Li, Liu, and Rosalsky (2010) on the asymptotic distribution of the largest entry Lₙ = ₁ ≤ ᵢ < ⱼ ≤ ₚₙ | ρ⁽ⁿ⁾ᵢ,ⱼ | of the sample correlation matrix Γₙ = (ρᵢ,ⱼ⁽ⁿ⁾)₁ ≤ ᵢ, ⱼ ≤ ₚₙ where ρ⁽ⁿ⁾ᵢ,ⱼ denotes the Pearson correlation coefficient between (X₁, ᵢ,..., Xₙ,ᵢ)' and (X₁, ⱼ,..., Xₙ,ⱼ)'. We show under the assumption EX² < ∞ that the following three statements are equivalent: align* & (1) ₙ → ∞ n² ∫(n n)¹/⁴∞ (Fⁿ⁻¹(x) - Fⁿ⁻¹(√n nx)) dF(x) = 0, & (2) (n/ n)¹/² Lₙ P→ 2, & (3) ₙ → ∞ P (n Lₙ² - aₙ ≤ t) = { - 1√8 π e⁻ᵗ/² }, - ∞ < t < ∞ align* where F(x) = P(|X| ≤ x), x ≥ 0 and aₙ = 4 pₙ - pₙ, n ≥ 2. To establish this result, we present six interesting new lemmas which may be beneficial to the further study of the sample correlation matrix.