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

Let { X , X k , i ; i ≥ 1 , k ≥ 1 } be a double array of nondegenerate i.i.d. random variables and let { p n ; n ≥ 1 } be a sequence of positive integers such that n / p n is bounded away from 0 and ∞ . This paper is devoted to the solution to an open problem posed in Li et al. (2010) [9] on the asymptotic distribution of the largest entry L n = max 1 ≤ i < j ≤ p n | ρ ˆ i , j ( n ) | of the sample correlation matrix Γ n = ( ρ ˆ i , j ( n ) ) 1 ≤ i , j ≤ p n where ρ ˆ i , j ( n ) denotes the Pearson correlation coefficient between ( X 1 , i , … , X n , i ) ′ and ( X 1 , j , … , X n , j ) ′ . We show under the assumption E X 2 < ∞ that the following three statements are equivalent: (1) lim n → ∞ n 2 ∫ ( n log n ) 1 / 4 ∞ ( F n − 1 ( x ) − F n − 1 ( n log n x ) ) d F ( x ) = 0 , (2) ( n log n ) 1 / 2 L n → P 2 , (3) lim n → ∞ P ( n L n 2 − a n ≤ t ) = exp { − 1 8 π e − t / 2 } , − ∞ < t < ∞ where F ( x ) = P ( | X | ≤ x ) , x ≥ 0 and a n = 4 log p n − log log p n , n ≥ 2 . To establish this result, we present six interesting new lemmas which may be of independent interest.