On a lemma of Butzer and Kirschfink

In [1], Butzer and Kirschfink discussed the convergence rates for C[0,1]-valued dependent random functions on Donsker's weak invariance principle and introduced the concept of dependency from below to deal with the martingale difference sequence. They asserted in their Lemma 8 that Lemma A. A martingale difference sequence (X_n,F_n,n≥1) with is dependent from below, i.e., for each 1≤i≤n and each n≥1 . The purpose of this note is to prove that Lemma A is not always true and to improve the conditions of Butzer and Kirschfink. We shall apply the notations in [1].