Maxima of increments of partial sums for certain subexponential distributions

We consider partial sums Sn=X1+X2+⋯+Xn, n∈N, of i.i.d. random variables with moments E(X1)=0, E(X12)=σ2 and sup{t∈R : E(exp((t|X1|)1/psgn(X1))<∞}∈(0,∞) and show thatlimn→∞max0⩽j<nmax1⩽k⩽n−jSj+k−Sjϕ(k/(log n)2p−1)(log n)p=1 a.s.with some explicit function ϕ(·). A related result for random variables with exponentially thin tails has recently been shown by Steinebach, extending a result given by Shao.