On the weak limit of the largest eigenvalue of a large dimensional sample covariance matrix

AbstractLet {wij}, i, j = 1, 2, …, be i.i.d. random variables and for each n let Mn = (1n) WnWnT, where Wn = (wij), i = 1, 2, …, p; j = 1, 2, …, n; p = p(n), and pn → y > 0 as n → ∞. The weak behavior of the largest eigenvalue of Mn is studied. The primary aim of the paper is to show that the largest eigenvalue converges in probability to a nonrandom quantity if and only if E(w11) = 0 and n4P(|ω11| ≥ n) = o(1), the limit being (1 + √y)2 E(w112)