On the limiting behavior of the Bahadur-Kiefer statistic for partial sums and renewal processes when the fourth moment does not exist (Q1185541)

\(X_ 1,X_ 2,\dots\) are i.i.d. random variables with positive mean \(u\) and finite variance. \(S_ n\) denotes \(X_ 1+\dots+X_ n\), and \(N(s)\) denotes \(\min[n\geq0;\;S_{n+1}>s]\). The strong limiting first-order behavior of the Bahadur-Kiefer-type statistic \(\sup_{0\leq s\leq n}| u^{-1}S_{[s]}+N(us)-2s|\) as \(n\) approaches infinity is investigated. It is shown that in the case where \(E(| X_ 1|^{4+d})\) is infinite for some \(d>0\), this behavior does not depend only on the mean and variance of \(X_ 1\), but on further characteristics of the distribution of \(X_ 1\). This is a different sort of behavior than for the case where \(E(| X_ 1|^ 4)\) is finite.