Strong and conditional invariance principles for samples attracted to stable laws

Abstract

We prove almost sure convergence of a representation of normalized partial sum processes of a sequence of i.i.d. random variables from the domain of attraction of an α-stable law, α < 2. We obtain an explicit form of the limit in terms of the LePage series representation of stable laws. One consequence of these results is a conditional invariance principle having applications to option pricing as well as to resampling by signs and permutations.