The index of the outstanding observation among n independent ones

Let X1, X2,... be independent random variables with distribution functions F1, F2,... respectively, Mn = max {X1,..., Xn} and Ln = min {k [less-than-or-equals, slant] n: Xk = Mn}. Assume that there exist constants an > 0 and bn such that (Mn - bn)/an converges in distribution to a non-degenerate random variable. It is easy to show that if for all t in (0,1) (M[nt] - bn)/an converges in distribution, so does Ln/n. We study the converse problem, namely, is it true that the convergence of Ln/n to a non-degenerate random variable implies the convergence in distribution of (M[nt] - bn)/an for all t[epsilon](0, 1)?