The law of the iterated logarithm for the last exit time of independent random sequences

Abstract

Let tβ=max{k ≧0:Xk>βk} if such a k exists and =0 else, be the last exit time of the sequence Xk of independent, identically distributed random variables with EXk+ 0. We will prove sufficient conditions such that the law of the iterated logarithm holds for tβ as β→0. In discussing the relationships to the maximum Zn=max{Xi, i≦n} we give weaker conditions for the law of the iterated logarithm of Zn(n→τ) than the known conditions.