Weak Convergence of Random Functions
نویسنده
چکیده
Let {vij}, i, j = 1, 2, . . . , be i.i.d. symmetric random variables with E(v 11) < ∞, and for each n let Mn = 1 sVnV T n , where Vn = (vij), i = 1, 2, . . . , n, j = 1, 2, . . . , s = s(n), and n/s → y > 0 as n → ∞. Denote by OnΛnO n the spectral decomposition of Mn. Define X ∈ D[0, 1] by Xn(t) = √ n 2 ∑[nt] i=1(y 2 i − 1 n), where (y1, y2, . . . , yn) = O (± 1 √ n ,± 1 √ n , . . . ,± 1 √ n ) . It is shown that Xn D −→ W ◦ as n → ∞, where W ◦ is Brownian bridge. This result sheds some light on the problem of describing the behavior of the eigenvectors of Mn for n large and for general v11.
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