A note on the largest eigenvalue of a large dimensional sample covariance matrix

Let {vij; i, j = 1, 2, …} be a family of i.i.d. random variables with E(v114) = ∞. For positive integers p, n with p = p(n) and pn → y > 0 as n → ∞, let Mn = (1n) Vn VnT , where Vn = (vij)1 ≤ i ≤ p, 1 ≤ j ≤ n, and let λmax(n) denote the largest eigenvalue of Mn. It is shown that limnλmax(n) = ∞ a.s. This result verifies the boundedness of E(v114) to be the weakest condition known to assure the almost sure convergence of λmax(n) for a class of sample covariance matrices.