We prove Strassen's law of the iterated logarithm for the Lorenz process assuming that the underlying distribution function F and its inverse F &1 are continuous, and the moment EX 2+= is finite for some =>0. Previous work in this area is based on assuming the existence of the density f :=F $ combined with further assumptions on F and f. 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. 1996 Academic Press, Inc.
1. INTRODUCTION AND THE MAIN RESULT
Let X be a non-negative random variable with distribution function F. We assume throughout that the mean + :=EX is finite and positive. The Lorenz curve corresponding to the random variable X, denoted by L F , is defined (cf. Gastwirth, 1971) by the formula
where F &1 denotes the left-continuous inverse of F.
In econometrics it is customary to interpret L F (t) as the proportion of total amount of wealth'' that is owned by the least fortunate t\_100 percent of a population.'' For some details on the variety of situations where estimating the curve L F is of importance, we may refer, for example, to: article no. 0050