On the limiting behavior of the Bahadur--Kiefer statistic for partial sums and renewal processes when the fourth moment does not exist

Let Sn = X1 + ... + Xn denote the nth partial sum of an i.i.d. sequence of random variables having positive mean [mu] and finite variance [sigma]2, and let N(s) = minlcubn [greater-or-equal, slanted] 0: Sn+1 > srcub denote the corresponding renewal process. We investigate the strong limiting first-order behavior of the Bahadur--Kiefer-type statistic defined by Dn = sup0[less-than-or-equals, slant]s[less-than-or-equals, slant]n [short parallel] [mu]-1S[s] + N([mu]s)-2s[short parallel] as n --> [infinity]. We show in the case where E([short parallel]X1[short parallel]4-[var epsilon])=[infinity] for some [var epsilon] > 0 that, unlike when E([short parallel]X1[short parallel]4) < [infinity], this behavior does not depend only upon [mu] and [sigma]2, but on further characteristics of the distribution of X1. In particular, our results imply that the assumptions of Theorem 3.2 of Horváth (1984) and of Theorem 1B of Deheuvels and Mason (1990) are sharp. This solves an open problem discussed in the latter paper.