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

Paul Deheuvels and Josef Steinebach

Statistics & Probability Letters, 1992, vol. 13, issue 3, 179-188

Abstract: 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)

Keywords: Bahadur; representation; partial; sums; and; renewal; processes; strong; laws; order; statistics; extreme; values (search for similar items in EconPapers)
Date: 1992
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