Strong limit of the extreme eigenvalues of a symmetrized auto-cross covariance matrix

The auto-cross covariance matrix is defined as Mn=12TT.j=1(eje.j+.+ej+.e.j), where ej.s are n-dimensional vectors of independent standard complex components with a common mean 0, variance .2, and uniformly bounded 2+.th moments and . is the lag. Jin et al. [Ann. Appl. Probab. 24 (2014) 1199.1225] has proved that the LSD of Mn exists uniquely and nonrandomly, and independent of . for all ..1. And in addition they gave an analytic expression of the LSD. As a continuation of Jin et al. [Ann. Appl. Probab. 24 (2014) 1199.1225], this paper proved that under the condition of uniformly bounded fourth moments, in any closed interval outside the support of the LSD, with probability 1 there will be no eigenvalues of Mn for all large n. As a consequence of the main theorem, the limits of the largest and smallest eigenvalue of Mn are also obtained.