Strassenʹs LIL for the Lorenz Curve

We prove Strassenʹs law of the iterated logarithm for the Lorenz process assuming that the underlying distribution functionFand its inverseF−1are continuous, and the momentEX2+εis finite for someε>0. Previous work in this area is based on assuming the existence of the densityf:=F′ combined with further assumptions onFandf. Being based only on continuity and moment assumptions, our method of proof is different from that used previously by others, and is mainly based on a limit theorem for the (general) integrated empirical difference process. The obtained result covers all those we are aware of on the LIL problem in this area.