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

Let {wij}, i, J = 1, 2, ..., be i.i.d. random variables and for each n let Mn = (1/n) WnWnT, where Wn = (wij), i = 1, 2, ..., p; j = 1, 2, ..., n; p = p(n), and p/n --> y > 0 as n --> [infinity]. 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([omega]11 >= n) = o(1), the limit being (1 + [radical sign]y)2 E(w112).