On weighted approximations in D [0, 1] with applications to self-normalized partial sum processes

摘要

Let X,X 1,X 2,… be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in D[0, 1] for the partial sum processes {S [nt], 0 ≦ t ≦ 1} where S n = Σ j=1 n X j , under the assumption that X belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes {S [nt]=V n , 0 ≦ t ≦ 1}, where V n 2 = Σ j=1 n X j 2. L p approximations of self-normalized partial sum processes are also discussed.