Equivalence of rates of convergence in the CLT between maximum partial sums and first passage times of random walks

Let {X n} by a sequence of iid r.v.'s such that {EX}1=µµ0 and Var X 1=σ2<∞. Let Sn = Σ n j=1 X j and šn=max1⩽κ⩽nSκ. Furthermore let N(c)=inf {k: Sk>C}, c⩾0. Set The following results are proved: 1. If Δ n =O(n−γ), 0<γ⩽½ then which is equivalent to w(c)=O(c−γ). 2. If then which is equivalent of